London Mathematical Society Lecture Note Series. 248
Tame Topology and O-minimal
Structures
Lou van den Dries
University of Illinois at Urbana-Champaign
.:. CAMBRIDGE
UNIVERSITY PRESS

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@ L. van den Dries 1998
This book is in copyright. Subject to statutory exception
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no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1998
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this book is available from the British Library
ISBN 0521 598389 paperback

CONTENTS
PREFACE vii
PREREQUISITIES viii
COSVESTIONS A:'\D KOTATIONS ix
ISTRODUCTION AND OVERVIEW 1
Chapter 1 SOME ELEMENTARY RESCLTS 11
1. Remarks on logical notation and boolean algebras 11
2. Elementary facts on structures 13
3. O-minimal structures 16
4. O-minimal ordered groups and rings 19
5. Model-theoretic structures 21
6. The simplest o-minimal structures 24
7. Sernilinear sets 25
Notes and comments 29
Chapter 2 SEMIALGEBRAIC SETS 31
1. Thorn's lemma and continuity of roots 31
2. Semialgebraic cell decomposition 33
3. Thorn's lemma with parameters 38
Notes and comments 41
Chapter 3 CELL DECOMPOSITION 43
1. The monotonicity theorem and the finiteness lemma 43
2. The cell decomposition theorem 49
3. Definable families 59
Notes and comments 61
Chapter 4 DEFINABLE INVARIANTS: 63
DIMENSION AND EULER CHARACTERISTIC
1. Dimension 63
2. Euler characteristic 69
Notes and comments 77
Chapter 5 THE VAPNIK-CHERVONESKIS PROPERTY IN 79
O-MINIMAL STR\:CTURES
1. A combinatorial dichotomy 79
2. Vapnik-Chervonenkis classes and dependence 81
3. Reduction to the case q = 1 85
f' otes an.d comments 91

vi
Chapter 6
Chapter 7
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
1. Curve selection
2. Fiberwise properties
3. Paths and partitions of unity
4. Curves, proper maps, and identifying maps
f' otes and comments
SMOOTHNESS
1. Differentiability in ordered fields
2. Inverse function thcorem
3. Definable maps are piecewise C 1
4. Existence of good directions
Notes and comments
93
93
98
100
102
106
107
107
109
114
117
118
Chapter 8 TRIANGCLATJON 119
1. Simplexes and complexes 119
2. Triangulation thcorem 127
3. Definable retractions and definable continuous extensions 134
Notes and comments 138
Chapter 9 TRIVIALIZATIO S 141
1. Trivialization theorem 142
2. Applications 149
3. On a conjecture of Benedetti and Risler 150
Notes and comments 154
Chapter 10 DEFINABLE SPACES AND QUOTIENTS 155
1. Definable spaces 156
2. Definable quotient spaces 161
Notes and comments 168
HINTS AND SOLUTIOSS 169
REFERESCES 173
INDEX 177

vii
PREFACE
My aim is to show that o-minimal structures provide an excellent framework for
developing tame topology, or topologie modirie, as outlined in Grothendieck's
prophetic "Esquisse d'un Programme" of 1984. This close connection between tame
topology and a subject created by model-theorists is hardly controversial, though
perhaps not yet widely known or understood.
In the early 1980s I had noticed that many properties of semialgebraic sets and
maps could be derived from a few simple axioms, essentially the axioms defining
"o-minimal structures", as their models came to be called in an influential article
by Pillay and Steinhorn. After Wilkie established in 1991 that the exponential
field of real numbers is a-minimal the subject has grown rapidly. The supply of 0-
minimal structures on the real field is still increasing. In combination with general
o-minimal finiteness theorems this gives rise to applications in real algebraic and
real analytic geometry.
A rough version of this book was circulated informally in 1991, and was based on
articles by various authors and on courses I gave at Stanford, Konstanz, and the
University of TIlinois at Urbana-Champaign. The main additions since then consist
of Chapter 5 about the Vapnik-Chervonenkis property, Section 2 of Chapter 6 on
fiberwise properties, Section 3 of Chapter 9 on a conjecture of Benedetti and Risler,
and Chapter 10 on definable spaces. I have taken pains to present the subject in
a way that is widely accessible and requires no knowledge of model theory. My
initial ambition was to include more on the construction of o-minimal structures,
but I decided to postpone this. (Traces remain as occasional references to what is
supposed to happen "in the next volume".) Some material is included explicitly
in the form of digressions or exercises requiring further background on the reader's
part. Also, in solving a problem of Benedetti and Risler I quote without proof
Wilkie's theorem of 1991. Otherwise everything is developed from scratch. Each
chapter has notes at the end with references and comments.
It is a pleasure to thank many people for their careful reading of parts of earlier
versions, their suggestions and their interest: M. Aschenbrenner, W. Henson, J.
Holly, J. Iovino, G. Kreisel, A. Lewenberg, A. Macintyre, D. Marker, C. Miller, A.
Nesin, S. Schanuel, P. Speissegger, S. Swierczkowski, A. Wilkie, and A. Woerheide.
The author is grateful for support by the National Science Foundation during the
years this book was in the making. I also thank J. Finkler for converting the text
into 'lEX.
May 1997
Lou van den Dries, Urbana, llIinois

viii
Prereq uisites
I tried to keep these minimal without compromising a natural development of the
subject. This book only requires familiarity with
From algebra. Basic properties of groups, rings (always associative with 1), fields,
polynomials, linear maps and their matrices and determinants.
From topology. The notions of (Hausdorff) topological space, closure and inte-
rior, continuity of functions between topological spaces, connectedness.
From analysis. Elementary calculus of real-valued functions of real arguments.
In Chapter 2 we use at one point the argument principle from complex analysis.
The logical dependence of chapters on earlier ones is roughly as follows:
1
,
i

2

3
I

4
j
6 5
:
1
7
j
8
A
9 10

ix
Conventions and Notations Used Throughout This Book
Our notations are fairly standard, but it may be useful to list some of them.
N, Z, Q, JR., and C denote the sets of natural numbers (= nonnegative integers),
intep;ers, rational numbers, real numbers, and complex numbers, respectively. We
always let k, m, n (possibly with accents or subscripts) range over N = {O, 1,2, . . .}.
AB
A-B
I:A-+B
. .. is by definition equal to . . . . . . j
A is a subset of the set B;
the relative complement {x E A: x rt B} of the set B in the set Aj
I is a map from the set A into the set B.
Formally, a map I: A -+ B is an ordered triple, cOll5isting of
(i) r(f):= {(a,/(a»: aEA} (the graph of I, a subset of Ax B),
(ii) the set A (the domain of f),
(iii) the set B (the codomain of f).
Given a map I : A -+ B and a set B '  B with I(A)  B ' we indicate the map
a ...... I( a) : A -+ B ' by I: A -+ B ' , and similarly, if B  B" we indicate the map
a ...... I(a) : A -+ B" by I: A -+ B". (This may be an abuse of notation but is
convenient and will not lead to confusion.) Also, with I as above and A'  A we
let IIA': A' -+ B be the restriction of I to A'. "Function" is used as a synonym for
"map". Usually we write "function" if we think of its values as numerical.
"Collection" is a synonym for "set"; we use "collection" if we think of its members as
sets themselves. We do maintain the distinction between a family (ai)iE/, which is
formally the assignment i ...... ai, and the set {as: i E I} of values ofthis assignment.
We let P( X) denote the power set of the set X, and let IXI be the cardinality of
the (usually finite) set X. (But we also write Ixl for the norm of a vector x, and
;KI for the polyhedron spanned by a complex 1(; precise definitions of Ixl and IKI
are given in the relevant chapters. We trust that this multiple use of the notation
I . I will not lead to confusion.)
Given a set R we let Rn be the set of all n-tuples (Xl,"" x,,) with all Xi E R.
Thus RO contains exactly one eJement, the empty tuple 0, so RO = {OJ.
If S  A x B and a E A we put Sa := {b E B: (a, b) E S}, and we view S as
describing the family (Sa)aEA of subsets of B. Usually this situation occurs where
a set R is given, and A = Rm, B = Rn, and A x B is identified with R m +".
A partition of a set S is a collection of nonempty pairwise disjoint subsets of S
whose union is S. A partition P of S is said to be compatible with a subset
X  5' (or is said to partition X) if X is a union of sets in P; in particular, P is
compatible with the empty subset of S. A refinement of a partition P of S is a
partition P' of S that is compatible with each set in P.

x
Let T be a topological space and S  T. If the ambient space T is clear from the
context then cl(S) indicates the closure of Sin T, and int(S) the interior of Sin T.
Also bd(S):= cl(S)-int(S) denotes the boundary of Sin T, and as := cl(S)-8
denotes the frontier of S in T. (Some books use other terminology or notation.)
If T is not clear from the context we write clT(S) and intT(S), We say that X
is an open (respectively, closed) subset of S if X is a subset of 5 and X is open
(respectively, closed) in the induced topology on S. Given a map f:8-+H from S
into a Hausdorff space II and given a point p E cl( S \ {p}) there is clearly at most
one point qEH such that for each neighborhood V of q there is a neighborhood U
of p such that f(U n (5 \ {p})) S;; V. We write
lim f(x) = q
.,.....1'
to indicate that q is such a point. (If in addition PES and f is continuous at p,
then this just means f(p) = q.)

INTRODUCTION AND OVERVIEW
Consider "nice" subsets of Euclidean spaces, defined by conditions like
(+) I1(x)=h(x)::=..'.=/.,(X)=0,91(X»0,'.',9'(X»0,
x = (Xl,"',X n ) ERn, for "nice" functions fl,...,fk,9l,...,9', Now perform
elementary logical, geometric and topological operations on these sets: take unions,
intersections, complements, closures and cartesian products, and project into lower-
dimensional euclidean spaces. If we keep repeating these operations with the new
sets that arise, then, roughly speaking, two kinds of things can happen:
(1) After only a few such operations, say after taking projections of finite unions
of the sets one starts out with, a stabilization occurs, and performing further
operations does not produce any new sets.
(2) Ever more complicated sets arise: Cantor-like sets, Borel sets of arbitrarily
high complexity, and so on. (For an example, see Chapter 1, (2.6).)
It is phenomenon (1) that is of particular interest to us: here we remain in the realm
of geometry and topology envisaged by Poincare, and we may view the study of the
corresponding category of "nice" spaces and maps between them as an unfolding
of the rich algebraic-analytic-topological structure of the continuum.
When phenomenon (2), on the other hand, is pursued, then the ties to geometry
tend to become very loose. We enter here Cantor's paradise: only set-theoretic fea-
tures ultimately survive in this bleak landscape. Already on the Borel level the real
line and the real plane (or Cantor space for that matter) become indistinguishable.
Despite these large contrasts it is often hard to determine for a specific class of
"nice functions" whether we are in case (1) or in case (2). In Chapter 1 we show
that with linear functions we are in case (1), and in Chapter 2 we do the same for
polynomial functions. But only in a later volume shall we pay special attention
to the art of proving that many interesting situations fall under phenomenon (1).
The emphasis in this book is on developing the general theory of what
else is true if one happens to be in case (1).
A key example of phenomenon (1) is provided by the semi algebraic sets. A semial-
gebraic set in R n is a finite union of sets ofthe form (*) where It, . . ., /k, 91, . . .,9' E
R[X], X = (Xl'" .,X n ). (If the polynomials /; and 9j are also assumed to be of
degree:::; 1 we speak of semilinear sets.) In particular, for n = 1, we get just
the finite unions of intervals and points, or equivalently, the subsets of the real line
with only finitely many connected components.

2
INTRODUCTIO AND OVERVIEW
A major fact about semialgebraic sets is the Tarski-Seidenberg projection property:
If AS;; 8."H is semialgebraic, then 1I'(A) S;; IR." is semialgebraic, where 11' :IR."H_IR."
is the projection map on the first n coordinates.
Related results are: the closure, interior, and convex hull of a semi algebraic set
are semialgebraic; a sernialgebraic set has only finitely many connected compo-
nents, each of which is also semialgebraicj a semi algebraic set can be triangulated,
and stratified into finitely many sernialgebraic real analytic manifolds. (All of this
remains true with "semilinear" instead of "semialgebraic".) A proper setting for
many results of this kind is provided by the tljeory of o-minimal structures, as this
book will attempt to show.
O-minimal structures on the real line
The class of semialgebraic sets and the class of semilinear sets, are both examples
of an o-minimal structure on IR., the ordered set of real numbers (the real line ).
DEFINITION. An o-minimal structure on R is a sequence S = (S,,) such that
for each n:
(1) S" is a boolean algebra of subsets ofR", that is, 8" is a collection ofsubsets
of IR.", 0e8" and if A, BeS", then Au BeS", and IR." - A eS"j
(2) AeS"  AxlR.eS"+landRxAeS"H;
(3) {(X1,...,X,,)elR.": xi==xj}eS"for1i<jnj
(4) AeS"+1  1I'(A)eS", where 1I':IR."H_IR." is the usual projection map;
(5) {r}eS1 for each relR., and {(x,y)eIR. 2 : x < y}eS2;
(6) the only sets in Sl are the finite unions of intervals and points. ("Interval"
always means "open interval", with infinite endpoints allowed.)
The first four axioms guarantee that each set A  IR." that is definable (in a precise
logical sense) from the sets in S also belongs to 8. (See Chapter 1.) With axiom
(5) they imply that finite unions of intervals and points belong to Sl. Axiom (6)
says that no further subsets of IR. belong to Sl; this is the minimality axiom that
explains the term "o-minimal", "0" standing for "order". This o-minimality axiom
can be viewed as expressing compatibility with the order (and hence topology) of
the real line, and is clearly the simplest compatibility condition of this nature.
Examples of a-minimal structures on the real line
. the semilinear sets;
. the semi algebraic sets;
. the subsets of the affine spaces IR." for n == 0,1,2,... that are subanalytic
in the larger projective space IP"(IR.);

INTRODUCTION AND OVERVIEW
3
. the images in R" for n = 0,1,2,... under projection maps R"+k -+ R"
of sets of the form {(x,y) E R"+k: P(x,y,e"',e) = o} where P is a
real polynomial in 2(n + k) variables, and where x = (Xt,...,x,,), y =
(Yt,...,Yk), eO: = (e"",...,e"'"), e'll = (e'll',...,e.).
These examples and others will be treated in some detail in this book and its
planned sequel: the first two in Chapters 1 and 2 of this book, the others in a
later volume. The o-minimality of the last example follows from a remarkable the-
orem proved by Wilkie in 1991. In contrast to the first three examples it contains
the graph of a function on R growing faster at 00 than any polynomial, namely eO:.
There are many other interesting o-minimal structures on R, but not too many. For
instance, a result of Peterzil [45] says that there is exactly one o-minimal structure
on R that lies strictly between the class of semilineal' sets ani! the class of semi aI-
gebraic sets; this intermediate structure contains every bounded semialgebraic set,
but not the graph of multiplication.
General facts about arbitrary o-minimal structures on the real line
From now on in this introduction we fix an o-minimal structure 8 on R.
Let A  R m and I:A-R". We say A is definable if AE8m. The map 1 is said
to be definable if its graph r(f)  Rm+" is definable. If 1 is definable, then the
domain A of 1 and its image I(A)  R" are also definable. The closure cl(A) and
interior int(A) of a definable set A  R" are definable. There are many more such
elementary but useful facts, and Chapter 1 is basically about them. A less easy but
fundamental result, proved in Chapter 3, is the following:
MONOTOSICITY THEOREM.
If the function f: (a, b) -+ R is definable, then there are a = au < at < '" < aN = b
such that 1 is continuous on each subinterval (ai, ai+t), and either constant, or
strictly increasing, or strictly decreasing, on each subinterval (ai, ai+d.
This result is used throughout the subject. In particular it plays a key role in
the proof that definable sets can be partitioned into finitely many cells. Cells are
nonempty definable sets of an especially simple form. They are defined inductively
as follows:
(i) the cells in Rt = R are just the points {r}, and the intervals (a, b)j
(ii) let C  R n be a cell; if I, g: C _ R are definable continuous functions such
that 1 < g on C, then
(f,g):= {(x,r)EC x R: I(x) < r < g(x)}
is a cell in R,,+1; a.lso, given a definable continuous function f: C -+ R the

4
INTRODUCTION AND OVERVIEW
sets
r(J),
(-00,1):= {(x,r)EG X JR.: r < I(x)},
(J,+oo):={(x,r)EGxJR.: I(x)<r},
are cells in JR.n+l; finally G X JR.  JR.n+l is a cell. (See figure.)
JR. t
I I
I I
I  ) ) J_ f(g)
;1 IT r rrrr(
I I I! ' : i _ (J,g)
I I I I I
i : II
\ \ ] I I .
II I l L  - r(f)
'-1 lJl JJ J :
G
Rn
It is extremely useful in inductive proofs to view the cells defined in (ii) as fiber
spaces over their projection G, and in general to view JR.nH as fibered over R n via
the projection map 11' : JR.n+l --+ R n on the first n coordinates. The next result is
also established in Chapter 3 and is central in the subject. It is to some extent the
many-variable version of the Monotonicity Theorem.
CELL DECOMPOSITION THEOREM.
Each definable set A  JR.m has a finite partition A = G 1 U' " U Gk into cells Gi. II
I: A -+ R n is a definable map, this partition 01 A can moreover be chosen such that
all restrictions flGi are continuous.
It follows in particular that a definable set has only finitely many connected com-
ponents, and these components are again definable (and path connected).

INTRODUCTION AND OVERVIEW
5
Definable invariants (Chapter 4)
The dimension of a cell is defined inductively in an obvious way, and then extended
to all definable sets A t:;; am by dimA:= max{dim(G): G t:;; A is a cell} if A is
nonempty, while dim(0) = -00. This dimension has good properties:
(1) Let I:A.....a" be a definable map, not necessarily continuous. Then
dim A 2': dim I(A); in particular dim A = dim I(A) if I is injective.
(2) dim(cl(A) - A) < dim A for nonempty definable A £;; am.
A more subtle "definable invariant" is the Euler characteristic. For a cell G of
dimension d it takes the value E( G) := ( _1)d, and for an arbitrary definable set A
we put
E(A) := L EeG.)
where G 1 ,. .., Gk are the members of a finite partition of A into cells. One proves
easily that this definition makes sense. More difficult is the following:
II I: A -+ R" is an injective definable map (not necessarily continuous!), then
E(A) = E(J(A)).
Results on definable families and definable collections (Chapters 4, 5)
Let S £;; a m +" be a definable set. Then 8 gives rise to a family (S",)", E]Rm of
definable subsets of a", where S",:= {yEa": (x, y)ES}. We call this a definable
family (with parameter space am), and we are interested in how the properties of
S", depend on x. Cell decomposition gives a uniform bound on the number of cells
into which 8", can be partitioned, independent of x, and a more precise analysis
along these lines leads to:
(1) For each dE {O,..., n} the set 8(d) := {x E am: dim(S",) = d} is definable
and
dim{(x,y)ES: dim(S",)=d} = dim(S(d))+d;
(2) E(S",) takes only finitely many values as x runs through R m . Moreover, lor
each integer e the set A(e) := {x ER m : E(Sx) = e} is definable, and
E{(x,y)ES: xEA(e)}:::: e.E(A(e)).
The collection of sets C := {Sx: x E am} is called a definable collection.
(We distinguish it from the family (8",)", e Rm.) This collection has a surprising
combinatorial property, as we explain now.

6
I:-TTRODUCTION AND OVERVIEW
For each finite set F  JR.", put FnC:= {Fn S,,: xEJR.m}, so 0  IFncl $ 2 1F1 .
Now define a growth function 9s: N -+ N by
9s(k):= max{iF n CI: F is a k-element subset of JR."},
so that 0  gs(k)  21:. A remarkable fact is:
(3) The function gs(k) is bounded by a polynomial in k.
(In probabilistic terms this means that {S,,: x E JR.m} is a VC-class or Vapnik-
Chervonenkis class; when R" is equipped with a probability measure it implies
that the law oflarge numbers holds uniformly for all sets Sz;, with bounds that are
independent of x.)
All the above actually goes through when instead of the real line R we take any
dense linearly ordered set without endpoints. The notion of o-minimal structure
on such an ordered set is defined in the same way as for JR.. (A basis for the
topology on the Of de red set is given by the intervals (a, b), with the product topology
on cartesian powers; the usual notion of "connected" is replaced by "definably
connected", which makes no difference when the underlying ordered set is R.) This
kind of generalization may seem at first unmotivated, but is actually crucial to prove
certain results in the classical real case. We discuss this later in the introduction.
The case that 8 contains addition and multiplication (Chapters 6-10)
Up till now we have considered a completely arbitrary o-minimal structure 8 on JR.,
and we did not even assume that 8 contains (the graph of) addition. Assume S
contajns addition. Then we have a very useful definable choice principle based on
the possibility of singling out the midpoint of a bounded interval. Special cases of
this genefal principle are the "curve selection lemma", and the fact that a definable
equivalence relation on a definable set has a definable set of representatives. These
facts have numerous consequences (see Chapter 6) but for deeper results we need
also the presence of multiplication. Assume S also contains multiplication.
Then in the cell decomposition theorem we can require the cells to be CI-manifolds,
and even C2- m anifolds, and so on (see Chapter 7). (However, for all I know it may
be that finer and finer partitions are required as the desired degree of smoothness
is increased.) We use CI-cell decomposition to prove a "good directions lemma":
Given a definable set A  R n+ 1 of dimension < n + 1 there is a unit vector u E sn
such that every affine line in JR. n+1 with direction u intersects A in only finitely
many points.
This is a substitute fOf Noether normalization in the proof of the following funda-
mental result in Chapter 8.

INTRODUCTION AND OVERVIEW
TRIANGULATION THEOREM. Each definable set is definably homeomorphic to a
bounded semilinear set.
One curious consequence of the triangulation theorem is that two definable sets
are definably equivalent (that is, there is a definable, not necessarily continuous,
bijection between them) if and only if they have the same dimension and the same
Euler characteristic. There are numerous other consequences, such as a definable
version of the Tietze extension lemma (Chapter 8, Section 3).
Call a definable map f: E ..... H between definable sets definably trivial if there
are a definable set F and a definable homeomorphism E ..... H x F such that the
diagram
E I BxF
/
B
commutes. (Note that then all fibers f-i(b) are definably homeomorphic to F.) In
Chapter 9 we prove
TRIVIALIZATION THEOREM. Let f: E --+ B be a definable continuous map between
definable sets E and B. Then H can be partitioned into definable sets Hi"", Hk
such that the restrictions
flr i (B;):r i (Hi).....B j , i = 1,.. .,k,
are definably trivial.
As a corollary, a definable collection {S",: x E R m} of definable sets contains only
finitely many definable homeomorphism types. Another standard consequence is
the conical structure of a definable set in some ball around any of its points.
In combination with the o-minimality of the exponential field of real numbers
(Wilkie's theorem) a variant of the trivialization theorem leads to the proof of
a conjecture by Benedetti and Risler (Chapter 9, Section 3).
In the final chapter, Chapter 10, we show that under rather weak assumptions
definable sets can be glued together by means of definable continuous transition
llIaps. We also use the triangulation theorem to form quotient spaces in the category
of definable sets and definable continuous maps.
The triangulation and trivialization theorems go through for each real closed field
equipped with an o-minimal structure that contains the addition and multiplication
of the field. In fact we prove these results directly in this setting, adapting the
treatment of the semialgebraic case by Delfs and Knebusch [11]. This kind of
generalization is done not for its own sake but to throw light on the classical real
case: certain real closed extensions of R, like the field of algebraic Puiseux series

8
INTRODUCTION AND OVERVIEW
over R, appear naturally as real closures of function fields of real algebraic varieties.
More generally, if R and a real closed extension R are equipped with corresponding
o-minimal structures, then points in Rn serve as generic points of definable subsets
of R.n. (This idea will only playa minor role here, but will become quite useful in
a later volume where the model-theoretic tools to handle such situations are made
available.) One could also argue that some real closed fields, like the field of real
algebraic numbers and, at the other extreme, Conway's field of surreal numbers,
are of independent interest. (In contrast to real closed fields in general, Conway's
field perhaps supports a canonical exponential function for which Wilkie's theorem
on exponentiation goes through.)
This finishes our preview ofthe main results. We actually prove.stronger and more
detailed versions of these theorems, and we also include exercises developing some
material a little further. To read this book only a rudimentary knowledge of real
analysis and point set topology is needed. In particular, no model theory is used,
although the subject of o-minimal structures developed in close contact with model
theory. The deeper results that will be treated in a later volume do require some
elementary results from model theory, valuation theory, and differential topology.
Notes, comments, and references
Artin and Schreier [1], Koopman and Brown [36] and Tarski [61] are at the origin
of much of our subject. Standard references for real algebraic and semi algebraic
geometry are the books by Brumfiel [7], Bochnak, Coste and Roy [4], and Benedetti
and Risler [2]; see also the articles by Bochnak and Efroymson [5], Delfs and Kneb-
usch [11], and Knebusch [34]. The theory of semianalytic and subanalytic sets was
created by Lojasiewicz [40], Gabrielov [26] and Hironaka [30]; see also Bierstone and
Milman [3], Denef and Van den Dries [15], and Sussmann [60], for expositions and
alternative treatments. Grothendieck's proposed topologie moderee in [28] would
contain all those theories as special cases and seems very much in the spirit of
o-minimality. The appendix in MacPherson [42] also puts forward a general view
on tame topology. It was observed in [20] that the theory of subanalytic sets falls
under the scope of o-minimalitYi see also Van den Dries and Miller [23] for more
on this, and for an exposition of o-minimality in the more geometric language of
manifolds.
Arbitrary o-minimal structures on the real line were first considered in [19], where
they were called "structures of finite type". The term "o-minimal structure" was
suggested by Pillay and Steinhorn [49], because of an analogy with the notion of
"strongly minimal structure" from model theory. This analogy has since turned
out to be very suggestive though it plays no role in this book. The article [49] and
its sequels [35] and [48] systematically develop the theory of o-minimal structures
on arbitrary linearly ordered sets from a model-theoretic viewpoint. This direction
found a recent culmination in Peterzil and Starchenko [46]. See also the survey
paper [21] for a description of the various roles of model theory in the study of
o-minimal structures. Chang & Keisler [9] is a basic reference for model theory.

INTRODUCTION AND OVERVIEW
9
An important event was Wilkie's proof in 1991 that the real exponential field is
model-complete and o-minimal, see [64]. This proof cleverly used model-theoretic
properties of o-minimal structures, and in return gave new insight into how to
establish model-completeness and o-minimality of certain structures on the real
field. A theorem of Khovanskii [32] on zerosets of exponential polynomials also
plays a key role in [64]. Khovanskii's book [33] elaborates on his theorem and
contains interesting applications and open problems.
For other o-minimal structures on the real field, see Van den Dries, Macintyre and
Marker [22] and Miller [44]. Building o-minimal structures on the real field has
become recently (1996-1997) a very lively activity: among the kinds of functions
that turn out to generate o-minimal structures are functions defined by Diriclflet
series and by multisummable power series, and Pfaffian functions. (I omit references
since at the time of writing much of this is only available in the form of preprints
and the dust has not settled yet.)
The following information came too late to include in the references at the end of
the book:
1. "Esquisse d'un Programme" has finally appeared (with English translation) in
Geometric Galois Actions. 1. Around Grothendieck's Esquisse d'un Programme,
edited by L. Schneps and P. Lochak, Cambridge University Press, 1997.
(Teissier's article there discusses tame topology.)
2. Tallie topology (in the real setting) is also developed extensively in the book
M. Shiota, Geometry of subanalytic and semialgebraic sets, Birkhauser, Boston,
Mass., 1997.

CHAPTER 1
SOME ELEMENTARY RESULTS
Introduction
The seven sections of this chapter are short and quite elementary. Sections 1, 2,
and 5 consist of generalities of a purely Jogical nature that are used throughout
this book. In Section 3 we formally introduce o-minimal structures and prove some
simple facts about them. In Section 4 we show that o-minimal ordered groups are
abelian and divisible, and that o-minimal ordered rings are real closed fields. In
Section 6 we prove the o-minimality of dense linearly ordered sets without endpoints
via a characterization of their "definable" sets. In Section 7 we do the same for
ordered vector spaces over ordered fields, and consider semilinear sets.
1. Remarks on logical notation and boolean algebras
(1.1) In this volume we will use logical formulas only in an informal and familiar
way, as convenient descriptions or definitions of sets and functions.
(1.2) To illustrate this notational use of formulas, let x, y, z be variables ranging
over nonempty sets X,Y,Z respectively, and let q'>(x,y,z) and (x,y,z) be condi-
tions on an arbitrary point (x,y,z)eX X Y X Z defining sets
  X x Y x Z and 'It  X X Y X Z,
:={(x,y,z)eXxYxz: q'>(x,y,z)holds},
'It:={(x,y,z)EXxYxZ: 1/J(x,y,z) holds}.
Then from the formulas <I>(x, y, z) and (x, y, z) we can construct new formulas:
q,(x, y, z) V 1/J(x, y, z), the disjunction of ljJ(x, y, z) and (x, y, z), defines  u 'ltj
q,(x, y, z) 1\ (x, y, z), the conjunction of </I(x, y, z) and (x, y, z), defines  n 'ltj
-,q'>(x, y, z), the negation of <I>(x, y, z), defines the complement (X X Y X Z) - ;
3zq'>(x, y, z), the existential quantification over z of q'>(x, y, z), defines the set
{(x,y)E;X x Y: there exists zEZ such that </I(x,y,z) holds},
11

12
SOME ELEMENTARY RESULTS
in other words, 3zq'>(x, V, z) defines the set 11'z( <p), where 11'z :XxYxZ --XxY is the
obvious projection map; similarly, the formulas 3xc?(x,V,z) and 3vljJ(x,V,z) define
the sets 11'x(<p) and 1!'y(<P) where 7rx:X X Yx Z -- Y x Z and 1!'y :Xx Y x Z --X xZ
are projection maps;
'r/zrl>(x,y,z), the universal quantification over z of rI>(x,y,z), defines the set
{( x, y) E X x Y: for all z E Z the condition 1jJ( x, y, z) holds}.
Note that this is also the set defined by the formula -.3z-.4>(x, y, z).
We write rI>(x, y, z) -- 'IjJ(x, y, z) as abbreviation for (-.IjJ(x, y, z») V 'IjJ(x, y, z), and
ljJ(x, y, Z)H'IjJ(X, y, z) to abbreviate (4)(x, y, z) -+'IjJ(x, y,z)) t\ ('IjJ(x, y, z)-- rI>(x, y, z)).
Instead of the symbol "/I" we also use the equivalent symbol "&".
As an example, the formula 'r/y(3zc?(x,y,z)-+3z'IjJ(x,y,z)) defines the set
{x EX: for all yEY such that there exists zEZ with r/1(x,y,z),
there exists z'EZ with 'IjJ(x,y,z')}.
(1.3) Of course, there is nothing special about the use of just three "underlying"
sets X,Y,Z. We may as well have more than three, or fewer, and it can happen
that X = Y, or Z = X', et cetera. L sually the variables range over the same
(nonempty) set, which is often clear from the context and not explicitly specified.
The main thing to keep in mind is that the various logical operations on formulas
correspond to operations on the sets these formulas define. We shali see in the
next two sections that logical notation is more suggestive and transparent than
traditional set-theoretic notation, in particular when quantifiers are involved.
(1.4) DEFINITION. A boolean algebra of Bubsets of a set X is a nonempty
collection C of subsets of X such that if A, BE C, then A u BE C and X - A E C.
Note that then X EC and 0EC, and that A, BEC implies An BEC.
An atom of a boolean algebra C of subsets of X is a minimal element (with respect
to inclusion) of {A E C: A oF 0} . For example, the atoms of the boolean algebra
'P( X) are exactly the singletons {x} with x EX. Given sets AI"", An <;;; X,
we denote by B(AI,"', An) the boolean algebra of subsets of X generated by
AI, . . ., An, that is, the smallest boolean algebra of subsets of X that contains
AI,' .., An as members. (The notation B(AI" ", An; X) would be more correct
since it indicates the ambient set X, but X will always be clear from the context.)
It is easy to see that the members of B(A I ,.. ., An) are exactly the finite unions of
sets of the form
(.) ( n Ai ) n ( n (X-A j ) ) ,with<;;;{I,...,n}.
i E.o- j f/..o-
Clearly the nonempty sets of the form (.) are the atoms of B(AI,...,An), so
B(AI"'" A,,) has at most 2" atoms, and hence at most 2 2 " members.

SOME ELEMENTARY RESULTS
13
2. Elementary facts on structures
(2.1) We define a structure on a nonempty set R to be a sequence 8 = (8 m )mEN
such that for each m > 0:
(51) 8m is a boolean algebra of subsets of R m ;
(52) if A E Sm, then R x A and A x R belong to Sm+!;
(53) {(XI,...,Xm)ER m : Xl =Xm}E8m;
(54) if A E 8 m +!, then 1I'(A) E 8m, where 11': Rm+!  Rm is the projection map
on the first m coordinates.
Instead of saying that 8 is a structure on R we also say that (R,8) is a structure.
The class of semi algebraic sets mentioned in the Introduction is an example of a
structure on 8., as will be proved in the next chapter.
Here is another important case. Let 11 be an algebraically closed field, for example,
11 = C, and let K be a subfield. The K -constructible subsets of 11 m are the finite
unions of sets of the form {x E 11 m : fl(X) = ... = fk(x) = 0, g(x) i: OJ, where
h(X),..., h(X),g(X) E K[X] and X = (Xl," .,X m ). That the K-constructible
sets form a structure on 11 is essentially Chevalley's constructibility theorem, also
known as quantifier elimination for algebraically closed fields. (In the next volume
we can give a very quick proof of this theorem. In this book we only use it as
illustration. )
The difficulty in proving in these cases that we are dealing with a structure is
the verification of axiom (54): closure under projections. In some other cases 8
is defined in such a way that this axiom is trivially satisfied, but then the diffi-
culty is usually located in verifying closure under complements. (In the next vol-
ume we will discuss some general logical ideas-quantifier elimination and model
completeness-that are often helpful in dealing with such problems.)
In the following lemmas we collect some elementary facts about a structure S = (Sm)
on R. First, we fix some terminology: a set A  R m is said to belong to 8 (or to
be in S) if it belongs to Sm; a map f:AB with A  R m and B  R" is said to
belong to S if its graph r(f) £; Rm+" belongs to Sm+,,'
(2.2) LEMMA.
(i) If AES m and BES", then A X BES m + n .
(ii) For 1 $ i < j $ m the diagonal Dojj := {(Xl,'''' X m ) E Rm: Xi = Xj}
belongs to 8.
(Hi) LetBES", andleti(l),...,i(n)E{l,...,m}. ThenthesetA£; Rm defined
by the condition
(Xl,.. .,Xm)EA  (Xi(1),.. "Xi("») EB
belongs to S. ("Permuting and identifying variables are allowed".)

14
SOME ELEMENTARY RESULTS
PROOF. For (i), note that A X B = (A x R n ) n (R m x B), and use (51) and
(52). For (ii), let II := {(X1,...,Xj_i) E Rj-i: Xl = Xj_i}, and note that
llij :::: R i - 1 X II X R m -H1, and use (53) and (i). For (iii) we note that
(Xl,..., x m ) E A  3Y1 '" 3Yn (X;(l) :::: VI & ,.. & Xi(n) = Yn & (Y1,''', Yn) E B).
Now think of (x, y) :::: (XI,"" X m , Y1,.", Yn) as ranging over Rm x Rn = Rm+nj
the formula "(Y1, . . . , Yn) E B" viewed as a condition on (x, V) defines the set R m x B
in 8, hence by (ii) the formula that follows the quantifiers 3Y1 . . .3Yn defines a set
in 8. Applying the quantifiers means taking successive images under the projection
maps
Rm+n -+ Rm+n-1 -+ . , ,-+ R m ,
hence AES m by (84). 0
(2.3) LEMMA. Let S  Rm and let f: S -+ Rn be a map that belongs to 8, that is,
r(f) E 8 m + n . Then we have the following properties:
(i) SES m ,
(ii) if A  S, AE8 m , then f(A)E8n and the restriction flA belongs to S,
(iii) if BE8n, then r 1 (B)E8m,
(iv) if f is injective, its inverse f-1 belongs to 8,
(v) if f( S) £;; T £;; R n , and g: T -+ RP is a second map belonging to S, then the
composition g 0 f: S --+ RP belongs to 8,
PROOF. Let X range over Rm and Y over Rn. For (i), use the equivalence xES 
3y(x, y) E r(f), for (ii) use the equivalence VE f(A)  3x(x E A & (x, V) H(f»),
for (iii) use
xEr 1 (B)  3Y(YEB & (x,V)Er(f»),
for (iv), use part (Hi) of the previous lemma, and the equivalence
(y,x)Ef (1-1)  (X,V)Er(f).
For (v), use the previous lemma, and the equivalence (with z ranging over RP)
(x,z)H(gof)  3V(x,V)Ef(f) & (y,z)Er(9»). 0
I hope these short arguments illustrate the utility of logical notation and its inter-
pretation in terms of set operations, There are numerous small facts of this kind
that the reader can verify instantaneously, by just writing down a suitable logical
formula. Below we give some exercises to become familiar with this technique,
which is practiced frequently but often implicitly in these notes. The results of
these exercises are also used in the rest of the book.
(2.4) Let R be a nonempty set. Given two structures 8(1) and 8(2) on R we
write 8(1) !:;; 8(2) if 8(1)m £;; 8(2)m for all mj this defines a partial ordering on

SOME ELEMENTARY RESULTS
15
the collection of structures on R. Any family (S(i)); e I of structures on R has a
greatest lower bound S in the collection of structures on R, namely
S = nS(i) with 8 m := n8(i)m, for each m.
i
(2.5) EXERCISES.
1. Let A t; R m and let I = (/t,..., In) : A  R n be a map with component
functions Ii: A  R. Show that I belongs to 8 if and ouly if each Ii belongs to 8.
2. (Sheaf property) Let'I be a finite index set, let AE Sm be the union of the sets
Ai E 8m (i E 1). Show that a map j : A  Rn belongs to 8 if and only if all its
restrictions jlA; belong to S.
3. Given A t; Rm+n and x E Rm we put Ar := {y E Rn: (x, y) E A}. Show that if
AE8 m + n and kEN, then the sets {XER m : IArl $ k} and {xER m : IAkl = k}
belong to Sm.
4. Let the sets A,B,C and the function I:A x BC belong to S. Show that the
set {a E A: I( a, .) : B  C is injective} belongs to S, and show also that the set
{aEA: I(a,.): B - C is surjective} belongs to 8.
5. Let P t; R be a non empty subset belonging to 8 1 . For m = 0,1,2,. .. put
(8IP)m := {An pm : AE8m},aboolean algebra of subsets of pm.
Show that SIP:= (SIP)m) is a structure on P. (The "restriction of 8 to P".)
6. Suppose 8 contains binary operations + : R 2 _ Rand . : R 2 - R with respect
to which R is a ring (always associative with 1 in this book). Show that S contains
{OJ and {I}, and that if S contains At; R m and the functions l,g:AR, then it
contains the functions -I, j + g, and I. 9 from A into R.
7. Suppose R = Rand 8 contains the order relation {(x,y)ER2: x < y}. Show
that the topological closure cI(A) of a set AES m also belongs to S. Show that if a
function j: R m+}  R belongs to 8, then the set
A:= {aER m : I(a,t) tends to a limit l(a)ERas t-+oo}
belongs to 8, and the limit function I:A-R so defined belongs to 8.
8. Suppose R = Rand S contains the graphs of addition and multiplication. Show
that S contains the order relation {(x, y) E R 2 : x < y}, and each singleton {q}
with q a rational number. Show that if 8 contains a function j: I  R, with open
I t; R, then it contains the set I' := {x E I: j is differentiable at x}, and the
derivative f': I' - R.

16
SOME ELEMENTARY RESULTS
(2.6) DIGRESSION: THE SMALLEST STRUCTURE ON THE REAL FIELD CONTAIN-
ING Z. In the introduction to this book I mentioned the possibility that, starting
with some relatively simple sets in euclidean space and performing the usual ele-
mentary operations on them, we obtain ever more complicated sets when we iterate
this generation process indefinitely. Here is the main example of this phenomenon.
Let S be the smallest structure on the set R that contains the graphs of addition
and multiplication on R, the subset Z of R, and each singleton {r}, r E R. Then
each Sn consists exactly of the so-called projective subsets of R n , in particular, all
Borel sets in Rn belong to Sn; see for example Exercise 37.6 in the book Classical
descriptive set theory by A. Kechris (Springer- Verlag, 1995).
The set Z of integers can of course be replaced here by the sine function on R, since
Z = {a E R : sin(1I'a) = OJ.
Those familiar with descriptive set theory and with Matijasevich's theorem on dio-
phantine sets may find it an amusing exercise to show that one can obtain all open
subsets of R n from a single polynomial equation over Z:
There is a polynomial Pn(X, Y, Z) with integer coefficients in the variables
X = (XO,X 1 ,.. .,X n ), Y = (YI,.. .,Y p ), Z = (Zl,', .,Zq) (for certain p,q E N
depending on n) such that {Ar : r E R} is equal to the collection of all open subsets
ofRn, where A  Rn+1 is the image of the set
{(x, y, z) E R n +1+p x zq : Pn(x, y, z) = O}
under the projection map Rn+1+PH  R n +1 onto the first n + 1 coordinates.
Once all open sets in euclidean spaces are available we also obtain their comple-
ments, the closed sets, hence all real-valued continuous functions on closed sets
(which have closed graphs), and so on. See Kechris's book referred to earlier for
how this generation process continues and never stops.
3. O-minimal structures
(3.1) NOTATIONS AND CONVENTIONS.
Given linearly ordered sets RI and Rz and a map f: RI  Rz we say:
f is strictly increasing if x < y in RI implies f(x) < fey) in Rz,
increasing if x  y in Rl implies f(x)  fey) in Rz,
strictly decreasing if x < y in R 1 implies f(x) > fey) in Rz,
decreasing if x  y in R 1 implies f(x)  fey) in Rz,
strictly monotone if f is either strictly increasing or strictly decreasing,
monotone if f is either increasing or decreasing.

SOME ELEMENTARY RESULTS
17
A linearly ordered set R is called dense if for all a, b E R with a < b there is c E R
with a < c < b. A subset X of a linearly ordered set R is called convex (in R) if
a < b < c with a,cEX implies bEX.
Let (R, <) be a dense linearly ordered non empty set without endpoints. (That
(R, <) has no endpoints means that R has no largest or smallest element.)
For convenience, we add two endpoints -00 and +00, with -00 < a < +00 for all
a E R, and put Roo := R U { -00, +oo}. An interval is always a nonempty "open"
interval
(a, b) := {XER: a<x<b}with -oo$a<b$+oo.
Note that intervals are convex.
Here are some further notations for certain kinds of convex subsets of R:
(a, b] .-
[a, b) .-
[a,b] .-
{XER: a<x$b}where -oo$a<b<+oo,
{XER: a$x<b}where -oo<a<bS+oo,
{XER: a$xSb}where -oo<a$b<+oo.
(N .B. Sets of these three kinds will never be referred to as intervals in this book.)
REMARK. We use the same notation for an interval (a, b) S;; R as for an ordered
pair (a, b) E R'. It should always be clear from the context which is meant.
We equip R with the interval topology (the intervals form a base), and each
product R m with the corresponding product topology, a base of which is formed
by the boxes in Rm: a box in Rm is a cartesian product (a1,bt) x .,. x (am,b m )
of intervals. Note that Rm is a Hausdorff space with this topology.
The (topological) closure in Rm of a set A S;; Rm is denoted by cI(A), and its
interior (in R m ) by int(A).
Given functions l,g:X -+R oo on a set X S;; R m we put
(I,g).- {(x,r)EXxR: I(x)<r<g(x)},
[/,g] .- {(x, r)E X x Roo: I(x) $ T $ g(x)}.
We consider (I, g) as a subset of Rm+1; also [/, g] S;; RmH if I and 9 are R-valued.
We write I < 9 to indicate that I(x) < g(x) for all XEX.
(3.2) Let (R, <) be a dense linearly ordered nonempty set without endpoints. An
o-minimal structure on (R, <) is by definition a structure 8 on R such that
(01) {(x,Y)ER 2 : x < V}E8"
(02) the sets in 01 are exactly the finite unions of intervals and points.

]s
SOME ELEMETARY RESULTS
In that case we also say that (R, <, S) is an o-minimal structure.
For the rest of this section we fix an o-minimal structure S on (R, <).
CONVENT10. Instead of saying that a set A belongs to S we usually follow more
traditional terminology and say that A is definable, and similarly with maps. This
is of course only permitted if it is clear from the context which S is meant.
The lemmas that follow are quite simple and will be used frequently without explicit
reference.
(3.3) LEMMA. Let A  R be definable. Then:
(i) inf(A) and sup(A) exist in Roo (Dedekind completeness for definable sets),
(ii) the boundary bd(A) := {x E R ; each interval containing x intersects both A
and R - A} is finite, and if al < . . . < ak are the points of b d( A) in order,
then each interval (ai, aiH), where ao = -00 and ak+1 = +00, is either
part of A or disjoint from A.
This is almost immediate from the definition of o-minimality. Here are other easy
results, the first one a variant of exercise 7 in (2.5) above.
(3.4) LEMMA.
(i) If A  Rm is definable, so are cl(A) and int(A).
(ii) If A  B  Rm are definable sets, and A is open in B, then there is a
definable open U  Rm with Un B = A.
PROOF. For (i), note the following equivalence:
(Xl,,' .,xm)E cI(A)

VY1",VymVzl",Vzm[(Yl<Xl<Zl & ... & Ym<Xm<zm)-+
3al.. .3a m (Yl <al <Zl & '" & Ym <am <zm & (all" .,am)EA)].
Now use the interpretation of the logical symbols in terms of operations on sets. For
(ii), note that one can take for U the union of all boxes in R m whose intersection
with B is contained in A. 0
REMARK. Lemma (3.4) uses only axiom (01) of o-minimality, and not (02).
Next we introduce a notion of connectedness appropriate for definable sets. In
Chapter 3, (2.19), exercise 7 we show that if the underlying ordered set (R, <) of
(R, <,8) is the ordered set of real numbers, then this agrees (for definable sets)
with the usual notion of connectedness.

SOME ELEMENTARY RESULTS
19
(3.5) DEFINITION. A set X  R m is called definably connected if X is definable
and X is not the union of two disjoint nonempty definable open subsets of X.
The verification of the following easy results is left as an exercise.
(3.6) LEMMA.
(1) The definably connected subsets of R are the following; the empty set, the
intervals, the sets [a, b) with -00 < a < b  +00, the sets (a, b] with
-00  a < b < +00, and the sets [a, b] with -00 < a  b < +00.
(2) The image of a definably connected set X  Rm under a definable continu-
ous map f: X  lln is definably connected.
(3) If X and Y are definable subsets of Rm, X  Y  cl(X), and X is definably
connected, then Y is definably connected.
(4) If X and Y are definably connected subsets of Rm and X n Y '" 0, then
X u Y is definably connected.
f'ote the following special case of (2).
If tIle function f: [a, b]  R is definable and continuous, then f assumes all values
between f(a) and f(b).
At this point it would be possible to continue directly with Chapters 2, 3, 4, and
5, since logically these depend only on the material just treated.
(3.7) EXERCISE. Let S  R m + n be definable. Show:
(i) {x E Rm : Sx is open} is definable.
(ii) {(x,y)ER m + n : YEint(Sx)} is definable.
4. O-minimal ordered groups and rings
(4.1) To formulate the next result we define an ordered group to be a group
equipped with a linear order that is invariant under left and right multiplication:
x < y =? zx < zy and xz < yz.
(So the additive group of reals, and the multiplicative group of positive reals, are
ordered groups under the usual ordering, but the multiplicative group of all nonzero
reals is not.)
(4.2) PROPOSITION. Suppose (R, <,8) is an o-minimal structure and 8 contains
a binary operation. on R, such that (R, <,.) is an ordered group. Then the group
(R,.) is abelian, divisible and torsion-free.
Denoting the unit element of the group by 1 we first prove

20
SOME ELEMENTARY RESULTS
LEMMA. The only definable subsets of R that are also subgroups are {I} and R.
PROOF. Given a definable subgroup G we first show that G is convex: if not, then
there are gEG, rER - G with 1 < r < g. This gives a sequence
1 < r < g < rg < g2 < rg 2 < l < . . .
whose terms alternate in being in and out ofthe definable set G, which is impossible.
So G is convex, hence assuming G:f {I} we have s:= sup(G) > 1 with (l,s)  G.
If s = +00, then clearly G = R. If s < +00, then we take any g E (1, s), and obtain
s = g . g-ls E G, since g-l s E (1, s), hence s < gs E G, contradicting the definition
of 8.
PROOF OF PROPOSITION (4.2). For each rER the centralizer C. := {xER: r:c =
:cr} is a definable subgroup containing r, so C. = R by the lemma. Hence R is
abelian. For each n > 0 the subgroup {x": x E R} is definable, hence equal to R.
This gives divisibility. Every ordered group is torsion-free. 0
(4.3) REMARK. Let (R, <, +) be an ordered abelian group, R :f {OJ, so (R, <)
has no endpoints. Assume also that the linearly ordered set (R, <) is dense. Then
the addition operation + : R2  R and the additive inverse operation - : R  R
are continuous with respect to the interval topology, that is, (R, +) is a topological
group with respect to the interval topology.
(4.4) In this book an ordered ring is a ring (associative with 1) equipped with a
linear order < such that
(i) 0 < 1;
(ii) < is translation invariant, that is: :c < y =? x + z < y + z;
(iii) < is invariant under multiplication by positive elements:
(x < y and z > 0 =? xz < yz).
Note that then the additive group of the ring is an ordered group in the previous
sense, that the ring has no zero divisors, that x 2  0 for all x, and that
k ...... k . 1: Z  ring
is a strictly increasing ring embedding, with the usual ordering on the ring Z of
integers; we shall always identify the ordered ring Z with its image in the ring under
this embedding.
Suppose our ordered ring is moreover a division ring: for each x :f 0 there is y
with x . y = 1. It is easy to check that such a y is unique (written as X-i), and
satisfies y . x = 1, and that x > 0 implies y > O. It is also easy to see that the
additive group is divisible, the underlying ordered set is dense without endpoints,

SOME ELEMENTARY RESULTS
21
and the maps (x, y) H xy and x H x-I (the last map defined only for x'" 0) are
continuous with respect to the interval topology.
(4.5) An ordered field is an ordered division ring with commutative multipli-
cation. Examples of ordered fields are the field of reals and the field of rational
numbers with the usual ordering. In this book we define a real closed field to
bc an ordered field such that if f(X) is a one-variable polynomial with coefficients
in the field and a < b are elements in the field with f(a) < 0 < f(b), then there
is c E (a, b) in the field with f(c) == O. (Intermediate value property.) So the or-
dercd field of reals is real closed and the ordered field of rational numbers is not
rcal closed. In the next volume we develop enough model theory so that we can
give an efficient and brief treatment of the theory o real closed fields of Artin
and Schreier, including Tarski's elimination theory (and much more). Here we will
rcstrict ourselves to the following:
(4.6) PROPOSITION. Suppose (R,<,S) is an o-minimalstructure and S contains
binary operations + : R 2 -+ Rand. : R 2 -+ R such that (R, <, +, .) is an ordered ring.
Then (R,<,+,.) is a real closed field.
PROOF. For each r E R we have a definable additive subgroup r R of (R, +), hence
rR == R if r '" 0, by the previous proposition. This shows that (R,<,+,.) is an
ordered division ring. Let Pos(R):== irE R: r > O}. Clearly Pos(R) is an ordered
multiplicative group. By restricting S to Pos(R) (see (2.5), exercise 5) it follows
from the previous proposition that multiplication is commutative on Pos(R), hence
on all of R. So (R, <, +,.) is an ordered field. By exercise 6 of (2.5), and the
remarks in (4.3) and (4.4), each one-variable polynomial f(X) E R[X] gives rise to
a definable continuous function x H f(:c) : R -+ R. Now apply the remark at the
end of (3.6) to this function. 0
5. Model-theoretic structures
(5.1) A structure S on a set R is often introduced by indicating relations and
functions on R that "generate" S. For example, in the next chapter we will see that
the structure on IR. generated by its addition and multiplication operations, and the
individual real numbers, consists exactly of the semi algebraic sets. It is quite useful
also for theoretical reasons to have available a notion of structure that singles out
particular relations and functions as "primitives".
(5.2) DEFINITION. A model-theoretic structure'R == (R,(Si)iEI,(fj)jEJ) con-
sists of a non empty set R, relations S;  Rm(i) (i E I, m(i) EN), and functions
fJ : Rn(j) -+ R (j E J, n(j) EN). If n(j) == 0, we identify f J with its unique value in
R, and call fj a constant. We also call R the underlying set of 'R, and the 8 i
and fJ the basic (or primitive) relations and functions of'R.
If the index sets I and J arc finite we usually just list the relations and functions.

22
SOME ELEMENTARY RESULTS
For example, an abelian group will be considered as a model-theoretic structure
of the form (A, 0, -, +), with 0 the zero element, - : A  A the operation of
taking the negative and + : A 2  A the group operation; an ordered ring is a
model-theoretic structure of the form (R, <,0,1, -, +,.), et cetera.
Given a model-theoretic structure 'R = (R, (Si)i El, (h)jE J), and set C S;; R we
define a new model-theoretic structure 'Rc := (R, (Si)i E I, (fj)jE J, (c)c E c) which
has the same underlying set and the same basic relations as 'R, but has in addition
to the basic functions h a constant c for each element c E C, where formally we
identify c with the corresponding function RO -+ R.
(5.3) Given a model-theoretic structure 'R = (R, (Si), (fJ)), we let Def('R) b the
smallest structure on the set R (in the sense of Section 2) that contains each relation
Si and each function fJ. Note that for each set C S;; R the structure Def('R c ) on
R is larger than or equal to Def('R). (We can recover Def('Rc) from Def('R) and
C in a simple way, see (5.9), exercise 1 below.) In this book we mostly work with
Def('RR)' The sets S t;:; R m and maps f: S -+ Rn that belong to Def('R) are also
called definable in 'R (or just definable, when 'R is clear from the context). A
point a = (al,"', am) E R m is said to be definable in 'R if the set {a} S;; Rm is
definable in 'R.
We usually write "definable in 'R using constants from C", or "definable in 'R with
parameters from C" in place of "definable in 'Rc". For C = R we write "definable
in 'R using constants" or "definable in 'R with parameters" instead of "definable iu
'RR'" We often omit "in 'R", if'R is clear from the context.
(5.4) To discuss in a uniform way all model-theoretic structures of a certain type,
for instance all ordered abelian groups, or all real closed fields, it is convenient to
have available the notions of a (first-order) language Land L-structure.
A language L is a disjoint union of two sets, a set of relation symbols, and a set
of function symbols, each relation symbol S and each function symbol f being
equipped with a number: arity(S), arity(f) E N. If arity(S) = m, arity(f) = n, W('
also say that S is an m-ary relation symbol, and f is an n-ary function symbol.
Function symbols of arity 0 are also called constant symbols, and instead of
saying "l-ary" and "2-ary" we say "unary" and "binary".
(5.5) An L-structure is a model-theoretic structure 'R = (R, (S1t), (J1t)) such
that for each m-ary relation symbol S and each n-ary function symbol f of L we
have
S1t S;; R m , f1t: R n  R;
S1t and f1t are the interpretations of the symbols 5' and f in 'R. So the set of
relation symbols of L indexes the primitive relations of 'R, and the set of function
symbols of L indexes the primitive functions of'R. We usually drop the superscript
'R in S1t and j1t, leaving it to the context whether the symbol or its interpretation
is meant.

SOME ELEMENTARY RESULTS
23
(5.6) EXAMPLES.
(1) The language ofabelian groups is Lab := {O, -, +}: it has a constant symbol
0, unary function symbol -, and binary function symbol +. We consider
abelian groups as Lab-structures in the obvious way.
(2) The language Lab( <) := {<, 0, -, +} of ordered abelian groups has in ad-
dition a binary relation symbol <, and we regard ordered abelian groups as
Lab( <)-structures in the obvious way.
(3) The language of rings is L ring := {O, 1, -, +, .}: it extends the language of
abelian groups by an extra constant symboll and an extra binary function
symbol.. We consider rings as Lring-structures in the obvious way.
(In a later volume we develop enough model theory to make better use of the notion
of L-structure. In this book it is only a matter of convenient terminology.)
(5.7) DEFINITION. A model-theoretic structure n = (R, <,...), where < is a
dense linear order without endpoints on R, is called a-minimal if Def(nR) is an
o-minimal structure on (R, <), in other words, every set S  R that is definable in
1? using constants is a union of finitely many intervals and points.
(5.8) EXPANSIONS. The simplest thing is just to say how we use this notion in
the few places that it occurs in this book. Let (R, <, S) be an o-minimal struc-
ture. We say that (R,<,S) expands an ordered abelian group if there are
abelian group operations 0: RO -+ R, - : R -+ R, and + : R2 -+ R belonging to S
such that (R, <,0, -, +) is an ordered abelian group; in that case we also say that
(R, <, S) expands the ordered Abelian group (R, <, 0, -, +). In particular, the
underlying ordered set of (R, <, S) is the same as the underlying ordered set of the
ordered abelian group that (R, <, S) expands. Similarly, we define what it means
for (R, <, S) to expand an ordered ring. Note that by the previous section, if the
o-minimal structure (R, <, S) expands a certain ordered ring, then this ordered ring
is act ually a real closed field.
(5.9) EXERCISES.
1. Let n = (R, . . .) be a model-theoretic structure, C  R, and A  Rn. Show that
A is definable in n using constants from C if and only if there exist a set S  Rm+n
definable in n, and elements Cl,' . ., C m E C such that for all (Xl,' . . , X n ) E Rn
(Xt,...,Xn)EA  (C\,...,cm,Xl,...,xn)ES.
. Let n = (R,. ..) be a model-theoretic structure and S  Rm+n definable in n.
Show that if aERm is definable in n, then the set Sa := {bERn: (a,b)ES} is
definable in n.
:I. Let n = (R,<,O,I,-,+,.) be the ordered field of real numbers. Show that
pach function! :R m -+R defined by a polynomia.l !(X l ,... ,X m ) E R[Xt,..., Xm]

24
SOME ELEMENTARY RESULTS
is definable in n using constants. Derive that each semialgebraic set in R m (see
"Introduction and Overview") is definable in n using constants.
6. The simplest o-minimal structures
(6.1) Let (R, <) be a dense linearly ordered nonempty set without endpoints.
We prove below that the model-theoretic structure (R, <) is o-minimal, by describ-
ing explicitly all definable sets. These o-minimal structures are too simple to be
of much interest, but we treat this case to illustrate the notions of the previous
sections. Starting with Chapter  we develop systematically the general theory
of o-minimal structures. However, in Chapters 6-10 we will assume that our 0-
minimal structures expand ordered abelian groups or even ordered rings. Such
extra assumptions are satisfied in all cases of real interest to us, but not by (R, <),
discussed in the present section. In this connection (R, <) can serve as a use-
ful counterexample to potential generalizations of various theorems in these later
chapters, showing that the extra assumptions in Chapters 6-10 cannot be omitted.
(6.2) Let 1  i  m. The function (Xl,,'" X m ) >-+ Xi: Rm -+ R will be denoted
just by Xi' The simple functions on Rm are by definition these coordinate functions
Xl, . . . , X m , and the constant functions Rm -+ R.
(6.3) Let /1," .,IN be simple functions on Rm, and let f:{I,.. .,NP-+{-I,O,I}
be given. Then we put
f(ft,. ..,IN):= {xER m : for all (i,j)E{I,...,N}2 we have
j;(x) < 'j(x) if f(i,j) = -1,
'i(x) = I;(x) if f(i,j) = 0,
li(x) > I;(x) if f(i,j) = I}.
Of course f(ft,..., IN) may be empty, for instance when (i,j) i: -f(j, i) for some
pair (i, j). Suppose that (ft,. . .,iN) is nonempty. Note that if  and 1) are the
restrictions of Ii and I; to (ft,.. ., IN), then either  < 1), or  = 1), or 1) < .
Let 1 < ... < k be the restrictions of ft,...,f N to ((11,"" 'N) arranged ill
increasing order. One checks easily that the sets r(j) (1  j  k) and the sets
(j,H1) (0  j  k, where o = -00 and k+l = +00 by convention) are exactl}
the nonempty subsets of Rm+1 of the form f'(/l,..., IN, xm+d, where
f':{I,...,N,N+ 1}2-+{-1,0,1}
is an extension of f. (Here a simple function Ion Rm is regarded as a simple functi()l
on Rm+1 by setting f(Xl'" ., X m , xm+tJ := f(X1'.'" x m ).) Define a simple set in
R m to be a su bset of Rm of the form f(ft, . . ., fN) with f1, .. ., iN simple funcUuIIs
on Rm and c {I,. ..,N}Z -+ {-1,0,1}. We have just proved that if S  Rm"l-l is
simple, then its image under the projection map
(Xl"", X m , xm+d H (Xl"", X m ) :R m + 1 -+R m
is simple in Rm. We now easily derive

SOME ELEMENTARY RESULTS
25
(6.4) PR.OPOSITION. The subsets of Rm that are definable in (R, <) using constants
are exactly the finite unions of simple sets in Rm.
PROOF. Let 8m be the collection of finite unions of simple sets in R m . Clearly 8m
is a boolean algebra of subsets of R m , and each set in 8m is definable in (R,<)
using constants. The considerations in (6.3) above show that 8 := (8m)m EN is a
structure on the set R, hence the sets in 8m are exactly the subsets of Rm definable
in (R, <) using constants. 0
REIARK. It follows easily that 8 is an o-minimal structure on (R, <).
(6.5) COROLLARY. The model-theoretic structure (R, <) is o-minimal.
(6.6) From a geometric viewpoint (R, <) can have rather bad properties. For
instance, given any cardinal I<. one can construct a dense linearly ordered nonempty
set (R, <) without endpoints and a collection C of I<. intervals in (R, <) such that
IIIi' IJI for distinct I, J E C. We leave the construction of such an (R, <) and
colJection C of intervals to the reader as an exercise, and note only the extreme lack
of homogeneity of such an (R, <) for large 1<.: C is a collection of I<. intervals no two
of which are homeomorphic. In Chapters 6-10 we impose extra algebraic structure
that implies homogeneity.
97. Semilinear sets
(7.1) In this section we show that the sets definable using constants in an ordered
vector space over an ordered field are exactly the semilinear sets. Ordered vector
spaces occur naturally in a variety of situations, but in this book the material of
this section will only playa minor role.
(7.2) DEFINITIOSS. Let F denote an ordered field, until further notice.
An ordered F-linear space is a vector space Rover F equipped with a linear
order making R an ordered additive group, such that for all >. E F and x E R
(>'>o,x>O) =? >.x>O.
Instead of "ordered F-linear space" we also say "ordered vector space over F". Of
course, F considered as a vector space over itself, with its usual order, is an ordered
P-linear space. More generally, any ordered field extension of F is an ordered F-
linear space in an obvious way. f'ote also that for F = Q the ordered Q-linear
paces are exactly the divisible ordered abelian groups, where the latter are made
lnto vector spaces over Q in the obvious way.
(7.3) Let R be an ordered F-linear space. We assume R '" {OJ, so that (R, <) is
a dense linearly ordered set without endpoints.

26
SOME ELEMENTARY RESULTS
An affine function on R m is a function f: R'" ..... R of the form
f(XI'" .,x",) = AIXI + ...+ AmXm + a,
where the Ai E F and a E R are fIXed. A basic semilinear set in Rm is a set of
the form
{xER m : l1(x) =... = fp(x) = 0, gl(x) > 0,. ..,gq(x) > O},
with the fi and gj affine functions on Rm. (We allow of course p = 0 or q = 0.)
A semilinear set in R m is a finite union of basic semilinear sets in R m .
(Perhaps "semiaffine" would be a better term than "semilinear", but "semilinear"
seems more popular.) ,
Note that the intersection of two basic semilinear sets in R m is a basic semilinear
set in R"'. The basic semilinear sets in R = R 1 are exactly the intervals, the points,
and the empty set. The complement in R m of a basic semilinear set in Rm is a
semilinear set in Rm, but not necessarily a basic semilinear set in Rm.
(7.4) PROPOSITION. Let II,..., fN be affine functions on Rm+l. Then we can par-
tition R m into basic semilinear sets C I , . .., C k such that for each C E {C!, . . . , C k }
there are functions C,I < . . . < C,j(C) from C into R with the following properties:
(i) each C,j is the restriction of an affine function on Rm;
(ii) each function fn (1  n  N) has constant sign on each of the sets r(c'J)
(1  j  j(C», and on each of the sets (c,j,C,Hl) (0  j  j(C), whcl'c
c,o := -00, C,j(C)+1 := +00 by convention).
PROOF. Let h,...,fM depend on the last variable x"'+1, while the fn for M <
n  N do not. The latter may therefore be viewed also as functions on R m . Writ(,
f p (xl,... ,X m , xm+d = Ap(gp(X!,... ,x",) - xm+d for 1  P  M, where Ap E F X
and Up is an affine function on Rm. Take a partition of R m into basic semilinea.r
sets C!,..., Ck on each of which each function in
{fM+l," .,fN} U {gp - Uq: 1  p, q  M}
takes constant sign. For each C E {C I ,..., Ck}, let C.l < ... < C,j(C) bc thr
restrictions of the functions gp to C arranged in increasing order. This is possihk
because the gp - gq are of constant sign on C. Then the desired result holds. Il
(7.5) REMARK. Note that each of the sets in (ii) is itself a basic semilinear set.
The actual construction in the proof produces sets as in (ii) that are of the forlll
{(x, t) EC X R: sign (fn(x, t») = £(n) for n = 1,..., N}
for a suitable sign description £: {1, . . ., N}..... { -1,0, I}. This property is somew b a:
stronger than (ii).

SOME ELEME:-TTARY RESULTS
27
(1.6) COROLLARY. Let 8m be the boolean algebra of semilinear subsets of Rm.
Then S := (Sm)m E N is an o-minimal structure on the ordered set (R, <). Each
function in S is piecewise affine, more precisely, given a function f: A..... R in 8 with
A  R m , there is a partition of A into basic semilinear sets Ai (1  i  k) such that
J I Ai is the restriction to Ai of an affine function on R m , for each i E {I, . . . , k} .
PROOF. Let 11': Rm+l -+ R m be the projection map onto the first m coordinates.
Given a set 5 E Sm+l, we have to show that 1I'(S) E Sm. Let It,. .., hr be the affine
functions inv01ved in a description of S as a union of basic semilinear sets. Applying
the proposition above to It, . . ., fN we see that 5 is a union of sets as described in
(ii). Since each of the sets in (ii) has as its 1I'-image a basic semilinear set in R m ,
it follows that 11'(5) E 8m. This shows that S is indeed an o-minimal structure on
(R, <). If 5 as above is the graph of a function f: A -+ R, A  Rm, then the sets
of type (ii) that are part of 5 must be of the form r(), with  a restriction of an
affine function. Hence f is piecewise affine. 0
(1.1) Construe R as a model-theoretic structure for the language
LF:={<,O,-,+}U{A': AEF}
of ordered abelian groups augmented by a unary function symbol A. for each A E F,
to be interpreted as multiplication by the scalar A.
(7.8) COROLLARY. The subsets of R m definable in the LF-structure R using con-
stants are exactly the semilinear sets in Rm.
PROOF. We leave it to the reader to derive this from (7.6). 0
(7.9) REMARK. The proofs above are "uniform in R", and one particular conse-
quence of this uniformity is that if R' is an ordered F-linear space extending R,
then an "elementary statement with constants from R" holds in R if and only if
it holds in R'. Since F can be embedded as an ordered F-linear space in each
nontrivial ordered F-linear space, it follows that all nontrivial ordered F-linear
spaces satisfy exactly the same elementary statements formulated in the language
L F . These issues will be rigorously treated in a later volume where the necessary
logical preliminaries are put in place. (For instance, we have to define "elementary
Slatement".) In this book we mainly deal with the situation that R = F, and we
then refer to the semilinear sets in R m for m = 0,1,2, . ., as semilinear sets over R.
(7.10) We never used the commutativity of the multiplication on F: everything
III this section goes through when F is an ordered division ring. ("Vector space
OVer the division ring}''' is then interpreted as "left F-module".) Noncommutative
ordered division rings can be obtained from noncommutative ordered groups via a
POWer series construction, see for example Fuchs [25].
(7.ll) An ordered vector space R is homogeneous in the sense that for each two
points p, q E R there is a definable homeomorphism from R onto itself that sends p

28
SOME ELEMENTARY RESULTS
to q, namely x H x + (q - p): R..... R. This homogeneity does not necessarily extend
from points to intervals, as the following example shows.
EXAMPLE. Equip the cartesian product IR. x Q of additive groups with the lexico-
graphic ordering:
(rl,/ll) < (rz,qz) iff rl < rz, or rl = rz and ql < qz.
This makes JR x Q into a divisible ordered abelian group in which there are both
countable intervals and uncountable intervals. In particular, not every two intervals
of IR. X Q are homeomorphic, let alone definably homeomorphic. This exa.mple is
used later to show that certain results on o-minimal structures expanding ordered
fields do not hold in general for o-minimal structures expanding ordered abelian
groups.
(7.12) DIGRESSION: THE DIVISION RING OF DEFINABLE ADDITIVE MAPS. We hav\'
just seen that ordered vector spaces over ordered fields, and, more generally, over
ordered division rings, are o-minimal. Here we show, using a result from Chapte,
3, that any o-minimal structure that expands an ordered abelian group is in fact
an ordered vector space over an ordered division ring that is canonically associatE'd
to the structure.
Let (R, <, S) be an o-mjnjrnal structure which expands an ordered abeUan group
(R,<,O,-,+).
Let A: R -+ R be a definable additive map. (Additive: A(X + y) = A(X) + A(Y).)
Since A is definablE', the monotonicity theorem from Chapter 3 implies that ,\ is
continuous at some point of R. Then additivity of A implies that ,\ is continuous
on all of R. Since R has no definable subgroups besides {O} and itself it follows
by looking at the kernel and the image of A that either A = 0 or A is a bijection.
Suppose ,\ is a bijection. Then (0,00) is the disjoint union of the two definable
open sets {x > 0: A(X) > O} and {x > 0: '\(x) < O}, so one of these two sets
is all of (0,00). In other words, either '\(x) > 0 for all x> 0, or A(X) < 0 for all
x > O. Write ,\ > 0 in the first case and ,\ < 0 in the second case. The set of
definable additive maps on R is a ring under pointwise addition and composition
with identity lR, and we have just defined an ordering on this ring. We have sho\\ n
PROPOSITION. The ring oj definable additive maps on R is an ordered divisinn
ring.
Hence R is naturally an ordered vector space over this division ring, and the srts
definable in this ordered vector space using constants are clearly also definable in
(R, <, S). There is more to say in this direction, see [21, 4].

SOME ELEMENTARY RESULTS
29
EXERCISE.
Let R be a nontrivial ordered vector space over an ordered division ring F, and
consider R as an LF-structure. Show that the maps R..... R that are additive and
definable using constants are exactly the scalar multiplications by elements of F.
(Hence these maps are actually dE'finable without using constants.)
Notes and comments
The results in Section 4 on o-minimal ordered groups and rings are from Pillay
and Steinhorn [49]. (There are other notions of "ordered ring" in use that allow
nonzero nilpotents, d. Fuchs [25].) The content of Section 6 is also known among
model-theorists as "elimination of quantifiers" for dense linearly ordered sets, cf.
Chang & Keisler [9]. The key argument in the proof of Proposition 7.4 is a form
of "Fourier-Motzkin elimination", and thus goes back a long time. See also Sontag
[55J.
The results in sections 4, 6 and 7 already hint towards a division of all o-minimal
structures into three fundamentally different kinds:
(1) Those in which very few sets are definable, like the dense linearly ordered
sets without endpoints.
(2) Non-trivial ordered vector spaces over ordered division rings.
(3) O-minimal expansions of real closed fields.
A precise result along these lines, the Trichotomy Theorem for o-minimalstructures,
has indeed been established by Peterzil and Starchenko [46]. This theorem can
be seen as a manifestation of the so-called Zil'ber Principle of geometric model-
theory, which clarifies the intrill6ic nature of such classifications. These matters fall
unfortunately outside the scope of this book.

CHAPTER 2
SEMIALGEBRAIC SETS
Introduction
In this chapter we prove that the sets definable with parameters in the field JR are
just the semi algebraic sets. This characterization of the definable sets is essen-
tially the Tarski-Seidenberg theorem. Here we give Lojasiewicz's proof, which gives
rather precise information on semi algebraic sets. The ideas involved will also be
useful when we deal with semianalytic sets and their generalizations in the next
volume. Lojasiewicz's proof was an eye-opener to me, and motivated the notions
of o-minimal structure and cell decomposition. Section 1 contains some lemmas
that are also useful elsewhere. In Section 2 we prove the main theorem (2.7), and
Section 3 characterizes closed semi algebraic sets.
1. Thorn's lemma and continuity of roots
We start with a simple but useful bound on the zeros of a polynomial with complex
coefficients:
(1.1) LEMMA. Let aEC be a zero of the monic polynomial
ao + alT +... + ad_lTd-1 + T d E C[T], d  1.
Then lal ::; 1 + max{la;l: i = 0,.. .,d - I}.
PROOF. Put M := max{;ad: i = 0,.. .,d - I} and suppose lal > 1 + M. Then
laO+ala+...+ad-1ad-11::; M'(1+lal+"'+laid-1) = M.(lald-l)/(lal-l) <
ICll d , contradicting 0 = if(a)1 = lao + ala + ... + ad-l a d - 1 + adl. 0
(1.2) THOM'S LEMMA. Let ft,...,hEJR[T] be nonzero polynomials such that il
II "'" 0, then It E {It, . .. , Id. Let c { 1, . . . , k} -t { -1 , 0, I}, and put
A.:={tEJR: sign(!;(t))=f(i), i=I,...,k},asubsetofJR.
Then A. is empty, a point, or an interval. II A. "'" 0, then its closure is given by
cl(A,) = {tEJR: sign (/;(t)) E{f(i),O}, i = 1,...,k}.
31

32
SEMI ALGEBRAIC SETS
If A, :::: 0, then {tER: sign (fi(t») E {(i), O}, i = 1,..., k} is empty or a point.
REMARK. We may call ( a sign condition for II,'", Ik. The 3 k possible sign
conditions (: determine 3 k disjoint sets A" which together cover the real line lR.. Of
course for many I:'S the corresponding A, may be empty. The second statement of
the lemma says that for nonempty A, its closure can be obtained by relaxing all
strict inequalities to weak inequalities.
PROOf OF THOM'S LEMMA. By induction on k. The lemma holds trivially for
k = O. Let fl,' .., /k, 1k+1 E R[TJ - {OJ be polynomials such that if II i 0, then
II E {f1, . . ., Ik+I}' We may as well assume that deg(!k+I) = max{ deg(f;): 1 
i  k + i}. l,et 1:' : {l,.. .,k + 1} -t {-1,0, I}, and let ( be the restriction of
1:' to {I,. . ., k}. By the inductive hypothesis A, is empty, a point, or an interval.
If A, is empty or a point, so is A., := A, n {t E lR.: sign (Jk+1 (t») = ('(k + I)},
and the other properties to be checked in this case follow easily from the inductive
hypothesis on A.. Suppose A, is an interval. Since 1+1 has a constant sign on
A., the function lk+1 is either strictly monotone on A" or constant. In both cases
it is routine to check that A,. = A. n {tElR.: sign(!k+1(t») ::::: ('(k + 1)} has the
required properties. 0
(1.3) LEMMA (CONTINUITY OF ROOTS). Let I(T) = ao + alT +... + adTd EC(T]
be a polynomial that has no zero on the ooundary circle Iz - c[ = r of a given open
disc Iz - cJ < T in the complex plane (c E C, r > 0). Then there is I: > 0 such that if
Ja; - bi! ::; I: for i = 0,.. .,d, then g(T):= b o + biT +... + bdTdEC[T] also has no
zero on the circle, and I and 9 have the same number 01 zeros in the disc. (Here
and in the proof below we count zeros wjth their multiplicity.)
PROOF. From complex analysis we know that the number of zeros of I in the disc
[z - cl < r equals
1 r I'(z)
21fi Jc fez) dz,
where the circle C is parametrized counterclockwise as c + re 21rl6 , 0  9  1. By
taking I: sufficiently small we can clearly guarantee that g has no zeros on C as well,
and then the number of zeros of g in the disc is given by the corresponding integral of
(1/21fi)(g'(z)lg(z»)dz over C. Moreover, for small enough (the difference [(1'1 f)-
(g'lg)1 will be 50 small everywhere on C that the values of the two integrals ovcr
C must differ by less than 1 in absolute value. Since these values are integers, they
must be cqual. 0
(1.4) EXERCISES.
1. Let I = I(X I ,. .., X m ) E F[X I ,..., XmJ be a polynomial with coefficients in
the field F, and let dl,...,d m E N be such that degx, I  d i for i = 1,...,m.
Show that if 1 vanishes identically on a cartesian product AI X '" x Am with
lAd> dl,...,IAml > d m (all A; S;; F), then 1=0.

SEMIALGEBRAIC SETS
33
2. Let F be an ordered field and f E F[X 1 ,. .., Xm], f i O. Show that the zero set
ZU) := {a E F m : f( a) = O} is a closed subset of F m with empty interior.
2. Semialgebraic cell decomposition
(2.1) We now come to the setting for Lojasiewicz's theorem:
X is a nonempty topological space, E a ring of continuous real-valued functions
f : X -t lR, the ring operations being pointwise addition and multiplication, with
multipicative identity the function on X that takes the constant value 1.
Call a set A S;; X an E-set if A is a finite union of sets of the form
{x EX : f(x) = 0, gl(X) > O,...,gk(X) > O}, with f,gl,...,gkEE.
l'i'ote that the E-sets form a boolean algebra of subsets of X.
(2.2) EXAMPLE. If X = lR m , and E = IR[Xb" ., xml is the ring ofreal polynomial
functions on X, then the E-sets are exactly the semialgebraic subsets of lR m.
(2.3) ote that the pair (X x lR, E[T]) also satisfies the conditions imposed on
(X, E), where X x lR is given the product topology, and each polynomial
f(T) = fo + hT +... + faTEE[TI
is interpreted as the (continuous) function (x, t) H f(x)t +...+ fo(x):X X lR-lR.
K OTATION. For f(T) as above and x EX, we write f( x, T) for the polynomial
f(x, T) = fo(x) + f1(x)T +... + /d(x)TElR[T].
(2.4) LEMMA. Suppose X is connected. Let f = fo + hT + .,. + fTd E E[T),
and suppose e  d is such that the polynomial f( x, T) E lR[T] has exactly e distinct
complex zeros, for each x EX. Then the number of distinct real zeros of f( x, T) is
also constant as x ranges over X. Writing (1 (x) < . . . < (r( x) for these real zeros,
the functions (j : X -t lR are continuous.
REMARK. Zeros are counted here without multiplicity.
PROOF. Let Xo E X and let Zl,..., z. be the distinct complex zeros of f(xo, T).
Take closed balls B i centered at Zj in the complex plane C, such that Bi n Bj = 0
for i f:. j, and Bi nlR = 0 if Zi lR. By "continuity of roots" there is a neighborhood
U of Xo in X such that for each x E U the ball B i contains at least one zero (i( x) of
f(x, T). Note that then (i(X) is in fact the only zero of f(x, T) in B i . The graph

34
SEMIALGEBRAIC SETS
of (j on U is {(x,t)E U X B j : f(x,t) = OJ, hence this graph is closed in U X Bi;
in combination with the compactness of Bj this implies that (i is continuous on U.
Since the coefficients of f(x,T) are real, the set {(l(X),.. .,(.(x)) is closed under
complex conjugation. Hence, if (i(XO) = Zi E JR, then (i(X) E JR for all x E U. This
shows that the number of real zeros is locally constant, and thus constant, since X
is connected. The real (i(X)'S must also keep their order on the real line as x runs
through U. 0
(2.5) The lemma below is a special case of Chevalley's constructibility theorem,
also known as quantifier elimination for algebraically closed fields, to be proved in
the next volume. We give here Lojasiewicz's direct proof of this special case.
(2.6) LEMMA. Let A = (Ao,..., Ad) be a tuple of distinct variables and let
f(A,T) = Ao + A1T +... + AdTdEZ[A,T]
be the general polynomial of degree d. Let e E {O,..., d} U {oo}. Then the set
{a = (aO,.. .,ad)EC d + 1 : f(a,T) has exactly e distinct complex zeros}
is a finite union of sets of the form
{aEC d +1 : Pl(a) =... = Pk(a) = O,q(a) ¥ OJ. where p;(A), q(A) EZ[A].
Note that for e = 0 the set is {(ao,O,...,O): ao ¥ OJ; for e = 00 the set is
just the one-point set {(O,O,...,O)}. This lemma implies in particular that for
f(T) = fo + f1T +.. '+fdTdEE[T] the set {xEX: f(x,T) has exactly e complcx
zeros} is an E-set.
PROOF OF (2.6). Let d> 0, and (ao,..., ad) E C d + 1 with ad ¥ O. Put 9 := degrec
of gcd(J(a, T), (&f /&T)(a, T») in C[T]j then the km ofthese two polynomials has
degree 2d - g-l, and the number of distinct complex zeros of f(a,T) is d - g. Lct
o < k < d. Hence the condition
(+)
{ f(a,T)' q(x,T) = (&f/&T)(a,T). r(x,T) for some nonzero
x = (xo,..., X2k) EC 2 k+l, where
q(x, T) = Xo + x1T + '" + Xk_1Tk-t and
r(x, T) = Xk + xk+tT + .,. + x2kT k
is equivalent to 2d - 9 - 1 :S; d + k -1, that is, to d - 9 :S; k, that is, to the condition
that f(a, T) has at most k distinct zeros. Clearly we have
f(a,T)' q(x,T) - (&f/&T)(a,T). r(x,T)
= ,Bo(a, x) + ,lh(a, x)T + ... + .Bd+k-t(a, x)Td+k-t

SEMIALGEBRAIC SETS
3S
for certain bilinear functions {3o,.. .,{3d+k-l :CdH X C'kH-+C.
Hence (*) is equivalent to the condition that {3i( a, x) = 0 for all i, and some nonzero
x E C'k+ l , that is, to the condition that the linear map
x...... ({3o(a,x),...,{3d+k_l(a,x)):C 2kH -+C dH
has nontrivial kernel. This last condition is equivalent to the vanishing of all
(2k + 1) X (2k + 1) minors of the matrix of this linear map. This exhibits the set
{a E C dH : ad f:. 0 and I( a, T) has at most k distinct complex zeros}
as the intersection of the set {a E C d +1 : ad f:. o} with the 'zero set of certain
polynomials in Z[A]. The desired result now follows easily. 0
(2.7) THEOREM. Let hCT),...,fM(T) E E[T]. Then the list f1,...,fM can be
augmented to a list f1,..., IN in E[T] (M  N), and X can be partitioned into
finitely many E-sets Xi (1  i  k) such that for each connected component G of
each Xi there are continuous real-valued functions C,l < . . . < C,m(C) on G with
the following two properties:
(1) each function fn (1  n  N) has constant sign (-1,0, or 1) on each of the
graphs r(C,j) (1  j  jl(G» and on each of the sets (c.i>c.i+d (0 
j  jl(C», where c.o := -00 and C,m(C)+1 := +00 are constant functions
on C by convention;
(2) each of the sets r(kj) and (c.j, c,i+d from (1) is of the form
{(x,t)EG x R: sign(Jn(x,t» = £(n) for n = 1,...,N}
for a suitable sign condition £: {1, . . ., N} -+ {-I, 0, 1}.
PROOF. Take dEN such that each of the polynomials fm(T) is of the form
fm(T) = fmo + fmlT + '" + fmdTd, with fmr E E for 0  r  d.
Let  range over the subsets of {1,..., M} X {O,. .., d}, and put
fA(T):= II (arlm/aT r ) EE[T], so degT(fA)  Md'.
(m,r) EA
We let e range over {O,. ..,Md',oo}, and put AAe := {x EX: fA(x,T) has
exactly e complex zeros}. From the remark following lemma (2.6) we obtain that
AAe is an E-set. Clearly, for a given  the sets AAe are pairwise disjoint as e varies,
and cover X. Because the E-sets form a boolean algebra we can find a partition
X == Xl U ... U Xk of X into E-sets Xi such that each set AAe is a union of Xi'S.
Augment f1,...,fM to fl,...,fN such that
{fl,...,fN} = {f}Tfm/8T r : lm$M, O$r$d}.

36
SEMIALGEBRAIC SETS
We shall prove that this sequence h,..., IN and these Xi'S satisfy the conclusion
of our theofem.
FOf each connected component C of an Xi, let fI.(C) be the set
{ (m, r) E {I,.. ., M} X {O,..., d} : :; does not vanish identically on C x lR} .
Fix such a component C. Given any fl., thefe exists e such that C £;; A.II.e, in
particular C is contained in A.II.(C)e for some e. Then !.o.(C) (x, T) has exactly
e complex zeros, for each x E C. We claim e is finite. To see this, considef a
factOf geT) = {}Tlm/OTr of I.II.(C), (m,r) E fI.(C). The set C is contained in
A{(m'.rne' fOf some e' E {O,. .., d, oo}, so the number e' of complex zeros of g(x, T)
is independent of x E C. Because (m, r) E fI.( C), this number e' is finite, and hence
e is finite. Lemma (2.4) now implies thefe are continuous real-valued functions
C,l < ... < C,;t(C) on C such that
{(x,t)ECxlR: 1.II.(c)(x,t)=O} = r(c,du...ur(k,..(c).
We shall prove that these functions c.j satisfy the conclusions of the theorem.
CLAIM 1. Each function or Im/OTr (1 :$ m :$ M, 0 :$ r :$ d) has constant sign
on each set rcC,j) (1 :$ j :$ J1.(C» and (c.j,c,j+d (0 :$ j :$ II(C», whefe
C.D = -00 and c.;t(c)+1 = +00.
Note that this claim gives us conclusion (1) of the theofem, since each In is one of
the or Im/8Tr>s. To pfove the claim, note fifst that or Im/OTr vanishes identically
on the indicated sets if (m, r)  fI.( C). Let (m, r) E fI.( C). Then, with geT) :=
or 1m /OT r , the number of complex zeros of g( x, T) is independent of x E C, as we
already saw. Hence by lemma (2.4) the zero set of g on C x lR is a finite disjoint
union of graphs of continuous real-valued functions on C. Since 1.II.(c) contains 9
as a factor, it is clear that these functions must be among the C,;8. (l'se that C
is connected.) This pfoves claim 1.
Now let B be one of the sets in claim 1. Put f(m, r):= sign(orl m /8T r ) on B,
and put
B':= {(X,t)EC x IR: sign ( ; ) (x,t) = t(m,r) for 1:$ m:$ M, 0:$ r:$ d}.
Clearly B £;; B'.
CLAIM 2. B = B'.
Suppose not. Take (x, t') EB' - B and (x, t)E B. Say t < t'. Thorn's lemma implies
that {SElR: (x,s)EB'} is connected, so {x} X [t,t'] £;; B'. Note that !.o.(C) must
change sign on {x} x [t,t'] since (x,t)EB and (x,t')B. But f.ll.(c) is a product of

SEMIALGEBRAIC SETS
37
partials (J"fm/8Tr, so fl!l.(c) cannot change sign on B', contradiction. This proves
claim 2, and thus conclusion (2) of the theorem. 0
(2.8) Let us say that the pair (X, E) has the Lojasiewicz property if each E-set
has only finitely many connected components, and each component is also an E-set.
(2.9) COROLLARY. If (X, E) has the Lojasiewicz property, then (X x R, E[T]) also
has the Lojasiewicz property, and the image of any E[T]-set !i  X X R under the
projection map X X JR - X is an E-set.
PROOF. Write an E[T]-set S as a finite union of sets of the form
{(x,t)EXxJR: f(x,t) =0, gl(X,t»O,...,gk(X,t»O}
with f, gl, . . ., gk E E[T]. Let It, . . ., fM be a list of all the polynomials in E[T] that
are involved in such a definition of !i. Now apply the theorem to this list. 0
(2.10) The R[T 1 ,.. ., T m]-sets in the space JR m are exactly the selnialgebraic subsets
of JR m. For m = 0 the space JR m has only one point, so the pair (JR m, JR[T 1 , . . . , T m])
obviously has the Lojasiewicz property for m = 0; hence by induction on m, using
(2.9), it follows that (JRm,JR[Tl,..., Tm]) has the Lojasiewicz property for each m,
and that the image of a semialgebraic subset of JR m+1 under the projection map
JR m+1_ JR m is a semialge braic subset of JR m. (Tarski-Seidenberg property)
We can now draw the following conclusion about the model-theoretic structure
(R, <,0, 1,+, -,.), which we will call "the ordered field of real numbers".
(2.11) COROLLARY. The sets S  JRm (m = 0,1,2,...) that are definable using
constants in the ordered field of real numbers are exactly the semialgebraic sets.
The ordered field of real numbers is o-minimal.
PROOF. By (2.10) the semialgebraic sets form a structure on (JR, <). Clearly the
primitives of the ordered field of reals are semialgebraic. By Chapter 1, (5.9),
exercise 3, the semialge braic sets are definable using constants in the ordered field
of real numbers. Finally, the semialgebraic subsets of It are clearly the finite unions
of intervals and points. The desired result follows. 0
This classical example should be kept in mind throughout this book. In the next
volume we extend this result by showing that all real closed fields are o-minimal, in
a uniform way, and indicate how this fact throws light on the classical real case as
well. There we also treat the remarkable recent theorem by Wilkie that the ordered
exponenUa] field of reals is o-minimal, and its aftermath.

38
SEMIALGEBRAIC SETS
3. Thom's lemma with parameters
This last section is mainly a preparation for our later work on semianalytic sets in
a later volume. We return to the set-up where we have a pair (X, E) as before: X
a nonempty topological space, E a ring of real-valued continuous functions on X.
(3.1) DEFINITIO;\,. Let f1(T),.. .,fM(T)EE[T], and let C be a connected subset
of X. A decomposition of f1, . . . , fM above C consists of continuous real. valued
functions 1 < .. . < r on C such that
(1) each function fm has constant sign on each set r(i) (1  i  r) and on
each set (i,'+1) (0  i  r" where o := -00, r+1 := +00);
(2) each set r(i) is contained in the zero set of some function fm'
Note: if (1) holds, then (2) can be satisfied by removing the i'S not satisfying (2),
Theorem (2.7) implies that if (X, E) has the Lojasiewicz property, then for every
sequence It, . . ., f M in E[T] there is a partition of X into finitely many connected
E-sets, above each of which f1, . . ., fM has a decomposition.
Let us say that a polynomial f(T) E E[T] has zero-free leading coefficient if
f(T) == cdTd + Cd-l T d - 1 + . . . + Co with Ci E E, such that Cd has no zero on X.
By (1.1) the real zeros of a polynomial adTd + ad_1Td-1 + '" + ao E lR[T] with
ad =I 0 are in absolute value  1 + max{ lai/adl: 0  i  d - I}.
Hence if f(T) E E[T] has zero-free leading coefficient and a point y E X is given,
then the real zeros of f( x, T) are bounded in absolute value by some constant, for
all x in a suitably small neighborhood of y.
This fact is used in the proof of the next lemma, which constitutes what one might
call "Thorn's lemma with parameters".
(3.2) LEMMA. Suppose the polynomials It,..., fN E E[T] have zero-free leading
coefficients, and each nonzero partial a fn/ aT also belongs to {It,. . ., fN}' Let
6 < ... < r be a decomposition of f1,. . ., f N above a connected set C  X. FiJ
i E{1,..., r}, let  = i, put c(n) = sign(Jnlr(O) forl  n  N. Then
(a) rw = {(x,t)EC x lR: sign(Jn(x,t)) = (n) for n = 1,.. .,N},
(b)  : C -+ lR extends uniquely to a continuous function Tf : cl( C) -+ It, and
cI(r(O) = r(Tf) = {(y, t) E cl(C) x lR: sign (In(Y, t)) E {((n),O} for n '"
1,...,N}.
P ROO F. For fixed x E C the set
A x ,,:= {tElR: sign (In(x, t)) = (n) for n = 1,... ,N}
contains {(x)} and is finite since some fm vanishes on r() and fm(x, T) =f O. Thr
polynomials It (x, T),..., fN(X, T) satisfy the hypothesis of Thorn's lemma, so A",,(
is connected. Hence Ax" = {(x)}. This proves (a).

SEMIALGEBRAIC SETS
39
For (b), take m such that Im(x,(X») = 0 for all x E C. Let Y E d(C). Since
Jm(T) has zero-free leading coefficient, there is a neighborhood U of y such that 
is bounded on Un C. Hence the limits
I .- liminf(x)
3H'!J
and L := limsup(x)
3H'!J
are real numbers (not ::1:(0). Now sign(Jn(x,(x»)) = fen) for all n, hence
sign (In(Y, I») E {(n), O}, and sign(fn(Y, L» E {fen), O} for all n. By Thorn's lemma
the set {t E lR: sign (In (y, t»)E {f( n), O} for n = 1,. .., N} is connected, so this set
contains [l,L]. With m as above we have Im(y,t) = 0 for all tE[/,L], and because
Jm(Y, T) f:. 0, this implies I = L. Since y was an arbitrary point of cl(C) we have
shown that 1](Y) := lim3H'!J (x) defines a function.,., : d( C) -t lR. Continuity of 
implies continuity of 1], so f(1]) is closed, hence cl(f({» = f(.,.,).
The argument with Thorn's lemma that we just gave also shows
q.,.,) ={ (y, t) E cl(C) x IR: sign (In(y, t»)E {e(n), O} for n = 1,.. ., N}.
This finishes the proof of (b). 0
(3.3) LEMMA. With the same assumptions aB in the previolJS lemma, set o := -00,
r+l := +00. Fix an i E {O,..., r} and put fen) = sign(Jnl(1t i+d). Then we
have
(a) (j,j....d = {(x, t)EC x lR: sign (In(x, t») = fen) for n = 1,..., N},
(b) cl(j,j+1) = {(y,t) E cl(C) x IR: sign(Jn(y,t») E {c(n),O} for n
1,...,N}.
We leave this as an exercise.
(3.4) FURTHER. TERMINOLOGY. Let F be a finite set of continuous real-valued
functions on a topological space X. We say that a set S  X is described by F
if S is a finite union of sets of the form
f(F) :={ x EX: sign(J(x» = f(f) for all IE F}, with cF -t {-I, 0, I}.
Since F is a finite collection of functions, the sets described by F form a finite
boolean algebra of subsets of X.
(3.5) NOETHER SORMALIZATION. Let I(X},..., X n ) be a nonzero polynomial of
degree d over a field K of characteristic 0, say I = E'il=d CiXi + terms of lower
degree, n > O. Here i = (iI, . .., in) E N n , i il = i l +. . +in and Xi = X;l . . . X", Cj E
K. Let (AI,.. ., An-d E Kn-l. We are going to make the (invertible) substitution
Xl -t Xl + AIX n ,
X n - l -+ X n - l + An-IX n ,
X n -+ Xn.

40
SEMIALGEBRAIC SETS
Note that, with Ai == Af' ... A:t', this substitution transforms f into
f(X t + AtX",... ,X n - t + An-1X n , X,,)
== ( L Ci Ai ) X + terms of degree < d in X".
lij=d
Since the polynomial 'E1il=d CiYl;' " . Y:l' E K[Y l , . . ., Yn-t] is nonzero, it follow
that for "almost all" A == (At,..., A,,-d E ](,,-t, namely for all those A that ar('
not zeros of this polynomial, the transformed polynomial
f(X l + A1X"" . ., X,,-l + An_tX",X,,)
is of (total) degree d, and has a term of the form cX with cE K - {O}.
(3.6) COROLLARY. A closed semialgebraic subset oj'R" is a finite union of sets of
thefQrm {x ERn: f1(:c) 2: O,...,fk(X)  o} with f1>...,fkER[Xt,...,X n ].
PROOF. By induction on n. The case n == 0 is trivial; assume the desired result
holds for a certain n, let S  R n +1 be a closed semi algebraic set, and let F be
a finite set of nonzero polynomials in R[X l ,.. ., X"' T] describing S. By a linear
transformation of variables as in (3.5) we may as well assume that all f E}' are
monic in T up to a constant factor, and hence, by enlarging F, we may also assume
that each nonzero partial {}r f / {}Tr of each f E F belongs to F. By theorem (2.7)
and by (2.10) there is a partition 'P ofR" into finitely many connected semialgebrair
sets C such that F has a decomposition over each C E 'P. !'low apply the inductive
assumption to the closed semialgebraic sets cI(C), and apply lemmas (3.2) and (3.:1)
to get the desired result. 0
(3.7) EXERCISES.
1. Let Q(X, Y) E R[X, Y] be a nonzero polynomial in two variables. Show that
there are dEN and M > 0 such that if (x, y) E lR 2 , X > M and Q( x, y) == 0, th(.n
Iyl  x d .
In the next two problems we say tha.t a map g:A.....lR n with A  Rm is semialge-
braic if its graph reg)  lR m + n is semialgebraic.
2. Show that if 9 : R ..... R is semialgebraic, then there are dEN and M > 0 such
that Ig(x)1  x d for all x > M.
3. Suppose a continuous function g:lRm.....R satisfies Q(x,g(x)) == 0 for all xEoR'"
and some nonzero polynomial Q(X, T) E R[X, T], X == (Xl,"" X m ). Show that
then 9 is semialgebraic.

SEMIALGEBRAIC SETS
41
Notes and comments
Classical sources for much of the material in this chapter are Koopman and Brown
'36], where analytic varieties are locally decomposed into analytic cells, Whitney
63], which stratifies real algebraic sets into finitely many connected semialgebraic
manifolds, and Lojasiewicz [40], where the Tarski-Seidenberg theorem is obtained
bv cell decomposition. See also the books by Benedetti and Risler [2; and by
Bochnak, Coste and Roy [4].
The proof of (3.6) given here follows the proof of a corresponding local result in
Lojasiewicz [40]. Other proofs are by Bochnak and Efroymson [5], Delzell [14],
Coste and Roy, see [4] j for a short model-theoretic proof, see (17].
The result of exercise 2 in (3.7) means that the ordered field of real numbers is
polynomially bounded. This property is shared by other o-minimal expansions
ofthis structure, but of course not by (IR, <,0,1, +, -,', exp). Miller [43] discovered
the surprising fact that every o-minimal expansion of the ordered field of reals
is either polynomially bounded, or defines the exponential function. Polynomial
boundedness implies the various Lojasiewicz inequalities, see [23J.

CHAPTER 3
CELL DECOMPOSITION
Introduction
In this chapter we establish two important results in the subject of o-minimality:
the monotonicity theorem (Section 1) and the cell decomposition theorem (Section
2). They are essential for everything that follows.
We work here with a fixed but arbitrary o-minimal structure (R, <, S).
111stead of saying that a set A  R m belongs to S we will say that A is definable,
a.5 is usual in the literature on o-minimal structures. Similarly with maps.
For further conventions and notations, see Chapter 1, Sections 1, 2,3.
l. The monotonicity theorem and the finiteness lemma
(1.1) The monotonicity theorem describes definable one-variable functions.
(1.2) MONOTONICITY THEOREM. Let f: (a, b) -t R be a definable function on the
interval (a, b). Then there are points a1 < ..' < ak in (a, b) such that on each
subinterval (aj,aj+1), with ao :;::: a, ak+1 :;::: b, the function is either constant, or
strictly monotone and continuous.
(1.3) We derive this from the three lemmas below. In these lemmas we consider a
definable function f: I -t R on an interval I.
Lf;IMA 1. There is a subinterval of I on which f is constant or injective.
LEMMA 2. If f is injective, then f is strictly monotone on a subinterval of I.
LEMMA 3. If f is strictly monotone, then f is continuous on a subinterval of I.
(1.4) These lemmas imply the monotonicity theorem as follows:
43

44
CELL DECOMPOSITION
Let
x {x E (a, b) : on some subinterval of (a, b) containing x the function f
is either constant, or strictly monotone and continuous}.
ow (a, b) -X must be finite, since otherwise it would contain an interval I; apply.
ing successively lemmas 1,2, and 3 we can make I so small that f is either constant,
or strictly monotone and continuous, on I. But then I  X, a contradiction.
Since (a, b) - X is finite, we can reduce the proof of the theorem to the case that
(a, b) = X, by replacing (a, b) by each of the finitely many intervals of which tlH'
open set X consists. In particular, we may assume that f is continuous. By splitting
up (a, b) further we can reduce to one of the following three cases.
CASE 1. For all xE(a,b), f is constant on some neighborhood ofx.
CASE 2. For all xE(a,b), f is strictly increasing on some neighborhood ofx.
CASE 3. For all x E (a, b), f is strictly decreasing on some neighborhood of x.
Case 1. Take Xo E (a, b) and put
s := sup{x: Xo < x < b, f is constant on [xo,x)}.
Then 8 = b, since s < b implies that f is constant on some neighborhood of .',
contradiction. From s = b it follows that f is constant on [xu, b). Similarly, WI'
prove that f is constant on (a,xo]. Therefore f is constant on (a, b).
Case 2. Take xoE(a,b) and put
8:= sup {x : Xu < x < b, f is strictly increasing on [xo,x)}.
Then 8 = b, since 8 < b leads to a contradiction as in case 1. Therefore f is strictly
increasing on [xo,b). Similarly, f is strictly increasing on (a,xo]. Hence f is strictly
increasing on (a, b).
Case 3. This is handled in the same way as case 2.
(1.5) We now prove the lemmas.
PROOF OF LEMMA 1. If some yER had infinite preimage f-l(y), then this preilll'
age would contain a subinterval of I and f would take the constant value y on tht
subinterval. So we may assume that each y E R has finite preimage. Then f( 1) j,
infinite, and so contains an interval J. Define an "inverse" g: J -+ I by
g(y):= min{xE!: f(x) = y}.

CELL DECOMPOSITION
45
Since 9 is injective by definition, g( J) is infinite, and hence g( J) contains a subin-
terval of I, and f is necessarily injective on this subinterval. 0
PROD F 0 F L EM MA 2. Let us write I = (a, b). We assume here that f is injective
and have to show that f is strictly monotone on some subinterval of I. For each
x E I the interval (a, x) is a disjoint union of two subsets,
(a,x) = {yE(a,x): f(y) < f(x)} U {YE(a,x): f(y) > f(x)},
so one of the parts contains an interval (c, x), a < c < x. The interval (x, b) breaks
up similarly. This shows that each x E 1 satisfies exactly one of the following four
for,Dmlas:
ct>++(x):= 3C!,C2E1 [Cl < x < C2 & 'v'yE(Cl'X): f(y) > f(x)
& 'v'yE(X,C2): f(y) > f(x)],
ct>+_(x):= 3Ct,C2E[ h < x < C2 & 'v'yE(Cl,X): f(y) > f(x)
& 'v'yE(X,C2): f(v) < f(x)],
and <lJ_+(x) and ct>__(x), which are defined similarly.
So 1 contains a subinterval all of whose points satisfy the same formula. Replacing
I by this subinterval, and f by its restriction to that subinterval, we may assume
that all points satisfy the same formula. This leads to four cases.
EASY CASE. ct>_+(x) for all x in I.
For each x in [ define s(x) := sup{s E (x,b): f> f(x) on (x,s]}. Then clearly
s(x) = b, since s(x) < b contradicts ct>_+(s(x)). Therefore f is strictly increasing
on I.
The case that c)+_(x) for all x in I leads similarly to the conclusion that f is
strictly decreasing on I.
DIFFICULT CASE. c)++(x) for all x in 1.
Let B := {x E 1: 'v'y E I (y > x implies f(y) > f(x))}. If B is infinite then B
contains an interval, and on this interval f is strictly increasing, and we are done.
So let us assume that B is finite. Passing to a subinterval to the right of all points
of B we may assume
(* )
'VxE1 3yE1 (y > x & f(y) < f(x)).
Let CE I. We claim that for all large enough y in 1 we have f(y) < f(c). Otherwise,
we would have f(y) > f(c) for all large enough YE (c, b). Take then dE [c, b) minimal
such that 'v'y (d < Y < b implies f(y) > f(c)).

46
CELL DECOMPOSITION
If fed) > fee) then d would not be minimal since .++(d). So fed) < f(c). But by
(*), there is e with d < e < band fee) < fed), so fee) < f(c), contradiction. This
proves the claim that fey) < f(c) for all large enough YEI.
Define y( c) as the least element of[c, b) for which f(y) < f( c) if y( e) < y < b. Not"
that iI>++(c) gives c < y(e) and f(y(c») < f(c). Minimality of y(c) clearly implies
that y(c) satisfies the following formula il'+_(v):
il' +_ (v) := 3Vi, V2 E I [Vi < V < v2 & 'v'Zl, Z2 (Vi < Zi < V < Z2 < V2 -+ f( zd > f(Z2»): .
Since c was arbitrary we have shown 'v'CEI 3veI (v> c & il'+_(v»).
Therefore il'+_(v) holds for all v in an interval of the form (d,b), dEl. Replacin
I by this subinterval we may as well assume that il'+_(v) holds on all of I.
A completely similar argument shows that we can pass to a still smaller subinterval
on which il'_+ (defined in the obvious way) holds. But this is a contradiction sine..
we cannot simultaneously have il'+_(v) and il'_+(v). The case that .__(x) hoJds
for all x in I is completely similar to the case we just handled. This finishes the
proof of lemma 2. 0
PROOF or LEMMA 3. Let us assume that the strictly monotone function f is strictly
increasing. (The case that f is strictly decreasing goes the same way.) Since f(1) i,
infinite there is an interval J  f(I). Take two points r, sE J, r < 8, and let c, d be
their preimages: f(c) = r, fed) = 8, C < d. Clearly f defines an order preserving
bijection of (c,d) onto (r,s). But the topology is defined in terms of the ord('1',
hence f is continuous on (c, d).
This finishes the proof of the lemmas and hence the proof of the monotonicity
theorem is complete. 0
Let us mention two easy but important consequences.
(1.6) COROLLARY 1. Let f: (a,b) -+ R be definable. Then for each c E (a,b) th,
limits limrTcf(x) and limr.Lcf(x) exist in R"",. Also the limits limrTbf(x) and
limr.Laf(x) exist in R"",.
COROLLARY 2. Let f : [a, b] -+ R be continuous and definable. Then f takes (l
maximum and a minimum value on [a, b].
Here is the other basic result of this section.
(1.7) FINITENESS LEMMA. Let A  R2 be definable and suppose that for each
x e R the fiber Ar := {yE R: (x, y) E A} is finite. Then there is N E N such thai
IArl ::; N for all xER.

CELL DECOMPOSITION
41
pROOF. A point (a, b) E R2 will be called normal if there is a box I x J around
(II, b) such that
either (I x J) n A = 0 (hence (a,b)IlA),
or (a, b) E A and (I x J) n A = f(J) for some continuous function 1: 1-> R.
(Note that in the latter case 1 is necessarily unique and definable.) Also, a point
(a, -00) E R x Roo is called normal if there is a box I x J disjoint from A such that
aEI and J = (-oo,b) for some b. Finally, (a,+oo)ER x Roo is called normal if
tbcre is a box I x J disjoint from A with a E I and J = (b, +00) for some b. Note
(1) The sets
{(a,b)ER 2 : (a,b) is normal},
{aER: (a,-oo) is normal},
{aER: (a, +00) is normal},
are definable.
1\ext we define functions f1, fz, ... ,In, ... by
dom(fn) .- {xER: IAxln},
and In (x) .- n tb element of Ax.
X otc that In is definable. (Its domain may of course be empty.)
Let a E R and take n  0 maximal such that f1, fz,..., In are defined and contin-
uous on an interval containing a. We call the point a good or bad, according to
whether
acI(dom(Jn+d)
a Ecl(dom(Jn+d)
- "good",
- "bad".
Let (; be the set of good points and B the set of bad points. Note that if a E (; then
(with n as above) the domain of In+1 is disjoint from an entire interval around a
on which 11, fz, . . . ,In are defined and continuous. This shows that for a E (; we
have
(2) !Axl is constant on an interval around a.
(3) (a, b) is normal for all bE Roo.
The key point of the proof is to show that Band (; are definable sets. (This is not
clear from their definitions which involve a parameter n E N depending on a.) To
this end we shall show
(4) If a E B then tlJere is a least bE Roo such that (a, b) is not normal.

48
CELL DECOMPOSITION
To see this, let us introduce for a E B the elements A(a, - ), A( a, 0), A( a, +) of Roo.
where n has the same meaning as before:
>.(a,-):= limfn+1(x) if fn+1 is defined on some interval (t,a),
xTa
.- +00 otherwise,
A(a,O):= fn+1(a) if aE dom(fn+d,
.- +00 otherwise,
A(a, +) := lim fn+1 (x) if fn+1 is defined on some interval (a, t),
xLa
:= +00 otherwise.
Now let ;3(a):= min{A(a,-), >.(a,O), >.(a,+)}. Then by checking the various
possibilities it is easy to see that ;3( a) is the least b E Roo such that (a, b) is not
normal. This proves claim (4), and together with (1) and (3) it implies that Band
9 are definable sets.
The rest of the proof is straightforward. Suppose first that B is finite, say
B = {al,.. .,ak}, with - 00 = au < al < ... < ak < ak+1 = +00.
We claim that IAxl is constant on each interval (ai, ai+d; just take any point a ill
this interval, and let n = IAal. Then by (2) the set {x E (ai, ai+d: IAxl = n} b
open, and for the same reason the set {x E (ai, ai+1) : IAxl 'f: n} is open. Since
both sets are definable the latter set must be empty.
Suppose now that B is not finite. We shall derive a contradiction from this assump-
tion, and that will finish the proof. Recall that {3( a) is, for a E B, the least bE Roo
such that (a, b) is not normal. Define the sets
B_:= {aEB: 3y (y < {3(a) & (a,Y)EA)),
B+:={aEB: 3y (y>{3(a) & (a,y) EA)},
and the functions {3- : B_ -+ R and {3+ : B+ -+ R by
{3_(a):= max{y: y < {3(a) & (a,y)EA},
{3+(a):= min{y: y> {3(a) & (a,y)EA}.
Since B is infinite by assumption, one of the sets B_ n B+, B_ - B+, B+ - B_,
B - (B_ U B+) is infinite, and each of these four cases leads to a contradiction. W<'
only show this in the case that B_ n B+ is infinite. (The other cases are similar.)
Since {3-, {3, and {3+ are definable functions, there is by the monotonicity theorem
aD interval I  B_ n B+ on which each of the functions {3-, {3, {3+ is continuom.
Note that {3- < {3 < {3+ on I. ow I splits into two subsets:
{xEI: (x,{3(x»EA},
and {xEI: (x,{3(x»A},

CELL DECOMPOSITION
49
and one of these subsets contains an interval. Replacing 1 by this subinterval we
may assume that either 1'(1311)  A, or 1'(1311) n A = 0. In either case it is clear
that r(i3!I) consists of normal points, since 13-, f3, f3+ are continuous on I. Now
we have a contradiction, since (a,13(a») is never normal. 0
(1. 8 ) We now combine the finiteness lemma with the monotonicity theorem in the
following result to be used in the next section.
Let A  R 2 be definable such that Ax is finite for each x E R. Then there are
points al < '" < ak in R such that the intersection of A with each vertical strip
(ai, a,+d X R has the form 1'(fi1) u ... u r(fin(i» for certain definable continuous
functions fij:(ai,ai+d-R with fi1(X) <... < fin(i)(X) forxE(ai,ai+d. (Here
we have set ao := -00, ak+1 := +00.)
(1.9) EXERCISES.
1. Suppose the function f: (a, b) - R on the interval (a, b) is definable. Show there
exist elements a1, . . . , ak with the property of the monotonicity theorem such that
al> .. . , ak are definable in the model-theoretic structure (R, <, 1'(f»).
2. Let / and J be intervals and f: I - R and 9 : J -> R strictly monotone definable
functions such that f(1)  g(J) and lim._r(I) f(8) = limt_r(J) g(t) in Roo, where
rU) and r(J) are the right endpoints of the intervals 1 and J in Roc. Show that
near these right endpoints f and 9 are reparametrizations of each other, that is,
there are subintervals I' of 1 and J' of J, with r(I) = r(I'), r(J) = r(J') and a
strictly increasing definable bijection h: l' -+ J' such that f(8) = g(h(s») for all
SE/'. (This will be used in Chapter 6, (4.3).)
2. The cell decomposition theorem
(2.1) In this section we show that a definable subset of Rm splits into finitely many
cens-definable sets of an especially simple form-and that each definable function
on a subset of Rm is "cellwise" continuous. For m = 1 this reduces for sets to the
definition of "o-minimal" and for functions to the mono tonicity theorem of the last
section.
(2.2) For each definable set X in Rm we put
C(X):= {J:X -R: f is definable and continuous},
C",,(X) := C(X) U {-oo, +oo},
where we regard -00 and +00 as constant functions on X.
For f, gin Coc(X) we write f < 9 if f(x) < g(x) for all x EX, and in this case we
put
(f,g)x:= {(x,r)EX x R: f(x) < r < g(x)}.

50
CELL DECOMPOSITION
So (1, g)x is a definable subset of RmH.
Usually we write just (1, g) instead of (f,g)x when X is clear from the context.
(2.3) DEFI:>ITION. Let (i l ,..., i m ) be a sequence of zeros and ones of length m.
An (il,''', im)-cell is a definable subset of R m obtained by induction on m as
follows:
(i) a (O)-cell is a one-element set {r}  R (a "point"), a (i)-cell is an interva:
(a,b)R;
(ii) suppose (il>..', im)-cells are already defined; then an (il,..., i m , O)-cell is
the graph f(1) of a function f E C(X), where X is an (il,''', im)-celJ:
further, an (ilt..., i m , i)-cell is a set (1,g)x where X is an (il"'" im)-cell
and f,gECoo(X), f < g. (See figure.)
Rt
I
!
I I
I I
I , r T  ) -- f(g)
 1 1 I , ' _
I I' ' I r r 1 1 r r  1 r
! I I: II I,
I I I I I i i I! : -- (1, g)
I I I ,I, I
" ,
, : , ' I
,,(, I:!I I I
\I.J.I I i jj ' l : tJ1 J 1-- r(f)
'l,Jl _ J :
,
I
,
j
)
R m
C
So a (O,O)-cell is a "point" {(r,s)}  R 2 , a (0, I)-cell is an "interval" on a vertical
line {a} x R, and a (1, O)-cell is the graph of a continuous definable function defined
on an interval. Note that a box in R m is a (1, . . ., 1 )-cell.
(2.4) TERMINOLOGY. A cell in R m is an (i l ,. .., im)-cell, for some (necessanly
unique) sequence (ii, . . . , i m ). Since the (1,.. ., 1 )-cells are exactly the cells wh if.1I
are open in their ambient space R m we call these open cells.
(2.5) The non-open cells are "thin";
The union of finitely many non-open cells in R m has empty interior. This is easily

CELL DECOMPOSITION
51
checked and we shall use this fact later on without further mention. Here is another
topological fact:
Each cell is locally closed, i.e. open in its closure.
To see this, let C <;; R m +1 be a cell. Put B := 1!'(C) <;; R m and assume inductively
that the cell B is open in its closure cI(B), so that cl(B) - B is a closed set. If
c::: f(J) with f: B....R a definable continuous function, then cI(C)-C is contained
in (cI(B) - B) x R, hence C is open in the closed set C U (( cI(B) - B) x R). If
C::: (f,g) with f,g:B-+R definable continuous functions on B, f < g, then one
easily verifies that cl(C) - C is contained in f(J) U f(g) U ((cl(B) - B) x R) and
that C is open in th closed set C U f(J) U f(g) U ((cl(B) - B) x R). (Draw a
picture.) The other cases are done similarly.
(2.6) COVENTION. We also consider the one-point space RO as a cell, more pre-
cise/y, as a ( )-cell, where ( ) is the sequence of length O. (It i8 an open cell; there
are no other ( )-cells.)
(In this way clause (i) in (2.3) appears as the case m = 0 of clause (ii). It allows
us to start inductions with m = 0.)
(2.7) Each cell is homeomorphic under a coordinate projection to an open cell. We
now make this explicit. Let i = (it,..., i m ) be a sequence of zeros and ones.
Define Pi : Rm .... R k as follows: let ),(1) < .. . < ).( k) be the indices ). E {I, . . ., m}
for which i>, = 1,80 that k = i i +... + i m ; then
Pi(Xi,...,Xm):= (x>,(1),,,,,X>'(k»)'
It is easy to show by induction on m that Pi maps each i-cell A homeomorphic ally
onto an open cell Pi(A) in R k . We denote Pi(A) also by p(A) and the homeomor-
phism pdA: A -+ p(A) by PA' Clearly PA = id A if A is an open cell.
(2.8) If A is a cell in Rm+1 then 1!'(A) is a cell in Rm, where 1!' :Rm+l..... Rm is the
projection on the first m coordinates. Here is a simple application of this fact.
(2.9) PROPOSITION. Each cell is definably connected.
PROOF. For intervals and points this is stated in Chapter 1, (3.6). If A is a cell in
R"'+1, then we assume inductively that the cell1!'(A) in R m is definably connected
and use the fact that each fiber 1!'-1 (x) n A is definably connected. 0
(2.10) DEFIl'OITION. A decomposition of R m is a special kind of partition of Rm
Into finitely many cells. The definition is by induction on m:
(i) a decomposition of R1 = R is a collection
{( -00, ad, (ai, a2),"" (ak, +(0), {ad,..., {ak}}

52
CELL DECOMPOSITION
where al < . . . < ak are points in Rj
(ii) a decomposition of RmH is a finite partition of RmH into cells A such that
the set of projections 1f(A) is a decomposition of R m . (Here 1f: RmH -+ Rm
is the usual projection map.)
Let V = {A(I),..., A(k)} be a decomposition of R m , A(i) :f; AU) if i =I j, and let
for each i E {I,..., k} functions f.l < .. . < f.n(.) in C(A.) be given.
Then
Vi := {( -00, fi! ),(f.l, f.2), .. ., (Jin(i), +00), r(f'I), .. ., r (fin(i»)}
is a partition of A(i) x R and one easily checks that V. := VI U ... U Vk is "
decomposition of Rm+l, and that every decomposition of Rm+1 arises in this way
from a decomposition V of Rm. We write V = 1f(V.). (See figure.)
R t :\ i
!
. Hf i 2) I


I ..
A(i) AU) R'"
A decomposition V of Rm is said to partition a set S  Rm if each cell in V is
either part of S or disjoint from S, in other words, if S is a union of cells in V. We
are now ready to state the main result of this chapter.
(2.11) CELL DECOMPOSITION THEOREM.
(1m) Given any definable sets AI,.." Ak  Rm there is a decomposition of R'"
partitioning each of AI, . . . , Ak'
(U m ) For each definable function f : A -+ R, A  Rm, there is a decompositior!
V of R m partitioning A such that the restriction fiB: B -+ R to each ell!
B E V with B  A is continuous.

CELL DECOMPOSITION
53
(2.12) The proof is by induction on m. ::-Jote that (Id holds by o-minimality, and
that (III) follows easily from the monotonicity theorem.
We now assume that (11)' '" , (1m) and (lId, ... , (II",) hold, and shall derive first
(I m +1) and then (II m +1)'
The proof is lengthy. The first step is to generalize the finiteness lemma of the
previous section. Call a set Y  R"'+1 finite over Rm if for each x E R m the fiber
y" : == {T E R: (x, T) E Y} is finite; call Y uniformly finite over R'" if there is
N E N such that IVxl :5:: N for all x E R"'. We shall use the finiteness lemma and
the inductive assumption to prove first
(2.13) LEMMA (UNIFORM FINITENESS PROPERTY). Suppose the definable subset
Y of Rm+l is finite over R m . Then Y is uniformly finite over R m .
PROOF. A box B  Rm will be called V-good if for each point (x, r) E Y with
x E B there is an interval I around T such that Y n (B x I) = f(J) for some
continuous function I:B.....R. (Note: this f is then uniquely determined by Y, B,
and I, and is definable.)
Claim 1. Suppose the box B  R'" is V-good; then there are continuous definable
functions h < ... < Ik in C(B) such that Y n (B x R) = f(h) U ... U f(Jk).
To see this, let us fix x E B and write Y x = {TI,..., rk} with TI < ... < Tk'
Take intervals It,..., Ik around rl,..', Tk respectively, and continuous functions
f1 , . .. ,Jk : B ..... R such that Y n (B x Ij) = f(Jj), j = 1,. .., k.
Subclaim a. h < . . . < Ik.
Let us prove only f1 < fz, the other inequalities following in the same way. Suppose
there is a point p E B wi th h (p) = fz (p). So 12 (p) E II, and by continuity of h there
is a neighborhood U  B of p such that fz (U)  II' Since Y n (U x II) = f(hl U)
and f(h IU)  Y n (u x It) it follows that It! U = fz IU. This argument shows
that the set {PEB: h(p) = f2(p)} is open. Since {PEB: f1(p) < h(p)} and
{PEB: 11 (p) > h(p)} are also open and B is definably connected, by (2.9), while
f1(x) = T1 < T2 = f2(x), it follows that 11 < fz.
Subclaim b. Y n (B X R) = f(fd U ... U f(lk).
Take any point (a, s) E Y n (B x R) and let I : B ..... R be a continuous definable
function such that I(a) = s and f(J)  Y. Since (x,J(x») E Y it follows that
I(x) = T; = f;(x) for some iE{I,.. .,k}. As in the proof of subclaim a we obtain
from this f = k This finishes the proof of claim 1.
A point x E R m will be called Y -good if x belongs to a Y -good box. Note that the
set of Y -good points is definable.
Clajm 2. If A  Rm is a definably connected set and all points of A are Y -good,

54
CELL DECOMPOSITION
then there are continuous functions It < .,. < fk in C(A) such that
Y n (A x R) = fUd U... U fUk).
To prove this we choose a point x E A, if A is nonempty, and let k = IYxl. By
claim 1 the set {a E A: IY. i = k} is open and closed in A, hence IY.I = k fOf all
a E A. Again, by claim 1, it is clear that It,. .., fk afe continuous.
Claim 3. Each open cell in Rm contains a Y-good point.
It is of course enough to show that each box B in R m contains a Y-good point.
Write
B == B ' x (a,b), B ' a box in R m - 1 .
For each point PEB' considef the set
Y(p):= {(r,B)ER 2 : a < r < b & (p,r,s)EY},
which is finite ovef R. Now we apply (1.8) to A = Y(p) and conclude that the set
{rER: r is not Y(p)-good} is finite. Therefore, the definable set
Bad(Y):= {(p,r)EB: r is not Y(p)-good}
has no interior point. By the inductive assumption (1m) there is a decomposition
of R m which paftitions Band Bad(Y). Take an open cell C of this partition such
that C £; B. Then en Bad(Y) = 0, so if we replace B by a box contained in C W('
have feduced to the case that Bad(Y) = 0, i.e., fOf each pEB' we can apply clailll
2 (with yep) £; R 2 instead of Y) to find a number k(p) E N such that IYxl = k}J)
for each point x = (p, r) E B. Next we have to show that there is a finite bound 011
the numbers k(p), p E B'.
To this end, we choose an rE(a,b) and considef the set
Y' := {(p,s): (p,r,s)EY} £; R m .
Since Y is finite over Rm, the set Y' is finite ovef Rm-t, so by the inducti\'('
assumption Y' is uniformly finite over Rm-l , i.e. thefe is N E N such that fOf ('aeh
pEB': l{sER: (p,s)Eyr}IN,thatis,l}(p.,)INforallpEB'. Helice
k(p)  N for all PEB'. Thus IYxl  N for all xEB.
For each i E {O,..., N} let Bi := {x E B: IYxl = i}, and define the function"
!it,... ,Jii on Bi by fit (x) < ... < fii(X), and Y x = {Jil(X),. .., /;i(X)}. AppIY:J:!(
the inductive assumption (U m ) to each fij separately, and then using (1m) to filld
a common refinement of the decompositions obtained via (U m ), we get a decompo-
sition 'D of R m partitioning each of the sets B i , such that for each A E 'D, if A k })"
then fijlA is continuous, j = O,...,i. Since B is open and is partitioned by P.
thefe is an open cell A E 'D with A £; B. Now B = U i Bi, so A k Bi for som" i,
therefore the functions fii, . . ., J;j are continuous on A. Hence each point of A is
Y-good. Since A  B this establishes claim 3.

CELL DECOMPOSITION
55
The proof of the lemma now proceeds as follows. Take a decomposition V of R m
partitioning the set of Y-good points. Let A eV. If A is open, then by claim 3 the
cell A contains a Y-good point, so all points of A are Y-good. By claim 2 there is
then a number N A E N such that IYxl $ N A for all x E A. By an easy exercise using
the definable homeomorphism PA (see (2.7)) such a number N A also exists for the
!lon-open cells AEV. Now take N:= max{N A : AEV}. Then IYxl :c:; N for all x
in R Tn . 0
(2.14) Before we proceed with the proof of (Im+d we recall from Chapter 1, (3.3)
hat a definable set S £; R has finite boundary bd(S), and that the interval between
tWO successive boundary points is either part of S or disjoint from S.
For a definable set A £; R m +1 we put
bdm(A):= {(x,r)ER m +1: rEbd(A x )},
alld we note that bd m (A) is a definable set which is finite over Rm, so that we can
apply the uniform finiteness property. This is the idea of the proof below.
(2.15) PROOF OF (Im+d. Let Ai,' ", Ak be definable subsets of R m + t . Put
Y := bdm(Ad U '" U bdm(Ak).
Then Y  R m +1 is definable and finite over Rm, so there is MEN such that
I }I :$ M for all x in Rm. For each i E {O, . . ., M} let B i := {x E Rm: IYx I = i},
aud define functions Iii, li2,...,1ii on Bi by
Y x = {Ji1(x),..., lii(x)}, /it (x) < ... < lii(x).
Further, put lio := -00, lii+1 := +00 (functions on Bi). Finally we define for each
AE{I,...,k}, iE{O,...,M} and 1:$j:$ i
Cij:= {XEB i : lij(x)E(A)x},
c.uu for each A E {I,..., k}, i E {O,..., M} and 0 :c:; j :$ i
Dij:= {XEBi: (Jij(X),/ij+l(X))  (A)x}.
We now take a decomposition V of R m which partitions each set Bi, each set Cij,
each set Dij, and which has moreover the following property:
if E eV is contained in Bi, then Iii IE, ..., liilE are continuous functions.
(Such a decomposition exists by the inductive assumptions (1m) and (IIm).)
For each cell E E V we let V E be the following partition of E x R:
V E := {(fioI E ,/iiI E ),. ", (fii IE,fii+1 IE), f(fidE),. '" f(fid E )},
where i E {O,. ..,M} is such that E  Bi. Then V. := U{V E : E E V} is a
decomposition of Rm+l which partitions each set At,. . . , Ak. 0
The proof of (II m +1) will be based on the following elementary lemma.

56
CELL DECOMPOSITION
(2.16) LEMMA. Let X be a topological space, (Rl,<), (R2,<) dense linear or.
de rings without endpoints and f : X x Rl -+ R2 U function such that for eu,'h
(x,r)EX x Rl
(i) f( x, .) : R 1 -+ R2 is continuous and monotone on R 1 ,
(ii) f(., r): X -+R 2 is continuous at x.
Then f is continuous.
PROOF. Let (x, r) E X X R 1 and f(x, r) E J, where J is an interval in R2' We shall
find a neighborhood U of x and an interval I around r such that feU x 1) £;; J. lIy
(i) there are r _, r+ in R 1 such that r _ < r < r+ and f(x, r_), f(x, r +) E J. Now lIe
(ii) to get a neighborhood U of:!: such that f(U x {r _}) £;; J and f(U x {r +}) £;; J.
We claim that then feU x 1) £;; J for I = (r_,r+).
Let x' E U and L < r' < r +. Assume f( x', .) is increasing. (The case that f( x', .) is
decreasing goes the same way.) Then f(x', r_) :$ f(x', r') :$ I(x', r +) and I(x', T _),
f(x',r+) are both in J, hence f(x',r') is in J. 0
(2.17) PROOF OF (II m +!). Let f: A -+ R be a definable function on a definable ,et
A£;; R m +1. We have to show that f is "cellwise" continuous. Because of (Im+l) ;t
suffices to show
(*)
{ A can be partitioned ino finitlY many efinable sets AI, . . . , Ak
such that flAi: Ai -+ R IS contmuous for z = 1, . . ., k.
Again because of (lm+l) the set A allows a partition into finitely many cells. So ill
order to prove (*) we may assume that A is already a cell. If the cell A is not open
in Rm+!, we use the definable homeomorphism PA: A-+p(A). Since peA) £;; R" roc
some n :$ m, it follows from the inductive assumption (IIn) that the set peA) raIL
be partitioned into definable sets Bl,. .., Bk such that (J 0 PAl )IBj is continuo]s
for each j. Hence A is partitioned into PA 1 (Bl)'" "PA1(Bk)' and the restriction
of f to each of these sets is continuous.
This establishes (*) for A non-open. Suppose now that A is an open cell. Call f
well-behaved at a point (p, r) E A if pEe for some box C £;; R m and a < r < b
for some a, b in R such that
(i) ex (a,b) is contained in A,
(ii) for all x E C the function f(x,.) is continuous and monotone on (a, b),
(iii) the function f(., r) is continuous at p.
Let A O be the set of all points of A at which f is well-behaved. ::-Jote that ;\' is
definable.
Claim. A ° is dense in A.
To prove this it suffices to show that, given any box B £;; Rm and -00 < a < (' <.
+00 such that B x (a, c) is contained in A, the box B x (a, c) intersects A o. fIj oW

CELL DECOMPOSITION
51
there is by the monotonicity theorem for each x E B a largest A(X) E (a, c] such that
the one-variable function f(x,.) is continuous and monotone on (a, A(X ». Since
'\: B --+ R is definable, there is by (IIm) a box C  B on which A is continuous.
Taking C small enough we may assume that b :$ A( x) for all x E C and a fixed
bE(a,c). Choose any element r in (a, b). The function f(.,r):C.....R is continuous
on some smaller box, by (IIm). Replacing C by this smaller box, we see that f is
well-behaved at each point (p, r) with pin C. This establishes the claim.
Now we take a decomposition V of Rm+l that partitions both A and A.. (Such a
decomposition exists by (lm+!)') Let DE V be any open cell contained in A.
We need only show that f is continuous on D. From D  A we obtain D  A",
since D intersects A" by the claim, and 'b partitions A.. In particular for each
point (p, r) in D the function 1(', r) is continuous at p. Therefore D is the union
of boxes C x (a, b) satisfying the conditions (i), (ii), and (iii) above, for each point
pEG, a < r < b. By lemma (2.16) the function f is continuous on each such box,
hence f is continuous on D. This concludes the proof of the cell decomposition
theorem. 0
We finish this section with an application of cell decomposition to definably con-
nected components, and some exercises.
A definably connected component of a nonempty definable set X  Rm is by
definition a maximal definably connected subset of X.
(2.18) PR.OPOSITION. Let X  R m be a nonempty definable set. Then X has only
finitely many definably connected components. They are open and closed in X and
form a finite partition of X.
PROOF. Let {Ct,..., Ck} be a partition of X into k disjoint cells. For each
nonempty set of indices [  {l,...,k}, put Cr := UiE/Cj. Among the 2 k -1
sets G r , let C' be maximal with respect to being definably connected. We claim
If a set Y  X is definably connected and C' n Y f:. 0, then Y  C'.
To see why, put Cy := U{ C; : C j nY f:. 0}. Since the C; 's cover X we have Y  Cy,
00 G y is the union of Y with certain cells that intersect Y. Hence Cy is definably
connected. But C' n Cy contains the nonempty set C' nY, so C' u C y is definably
connected. By maximality of C' it follows that C' u Cy = C/. Hence Y  C y  C',
which proves the claim. It follows in particular that C' is a definably connected
component of X. It also follows that the sets of the form C' form a (finite) partition
of X. Furt her the claim shows that the sets C' are the only definably connected
components of X. Note that because the closure in X of a definably connected
subset of X is also definably connected, the definably connected components of X
<ire closed in X. Hence they are also open in X. 0

58
CELL DECOMPOSITION
(2.19) EXERCISES.
In the first four exercises we strengthen the cell decomposition theofem in several
ways. This requires a somewhat technical notion of fegularity:
(i) Call an open cell C £; R m regular if for each i E {I,. . ., m} and each two
points x, Y E C that differ only in the i 1h coordinate and each point z E IC"
that differs from x and y only in the i 1h coordinate, we have Xi < Zi < Yi =}
ZEC. (Example: boxes in Rm are regular.)
(ii) Consider a definable function f: C --+R, where C £; R m is a regulaf open ct'li.
Call f strictly increasing in the i th coordinate (1 ::; i :::; m) if fOf all
points x, Y in C that differ only in the i 1h coordinate, with Xi < Yi, we hay"
f( x) < f(y); similarly we define the notions of "f is strictly decreasing
in the i th coordinate" and "I is independent of the i th coordinate",
the latter meaning that f(x) = fey) whenever x,y E C differ only in the
i 1h coordinate. Finally, let us say that f is regular if f is continuous, alid
for each i E {I,..., m} the function 1 is eithef stfictly increasing in the i th
coordinate, or strictly decreasing in the i 1h coordinate, or independent of
the i th coordinate. (Which of these three cases takes holds may depend on
i. )
1. Let C £; R m be a regulaf open cell and I: C --+ R a regular definable function.
Show that the open cells (-oo,!), (/,+00) and C x R = (-00,+00) in RmH Ml'
regular. Show that if 9 : C --+ R is a second regular definable function with f < g.
then the open cell (I,g) in Rm+l is regular.
2. Prove by induction on m the regular cell decomposition theorem:
(Im) For any definable sets Ai,..., Ak  Rm there is a decomposition of urn
partitioning each Ai, all of whose open cells are regular.
(IIm) For each definable function f: A..... R, A  R m , there is a decomposition J1
of Rm partitioning A all of whose open cells are regular, and such that for
each open cell C E V with C £; A the restriction flC is regular.
3. Let C be a cell in R m , D = (Q, [j)c a cell in R m +1 and f: D --+ R a defina hlr
function such that for all x E C the function f( x, .) : (o( x),,B( x») --+ R is continuoll'.
Show that C can be partitioned into cells Cl"..,C k such that, with Qi:= 0:1('"
,B; := ,BIC i , each restriction fl( Qi',6;): (Qi,,8;) --+ R is continuous.
4. Improve the cell decomposition theorem as follows:
(1m) If the sets Ai, . . ., Ak £; Rm are definable, then there is a decompositioll of
Rm partitioning each set Ai, all of whose cells are definable in the modi'!
theoretic structufe (R, <, Al,..., Ak)'
(IIm) Let the function f: A --+ R, A £; R m , be definable. Then there is a de'
composition V of Rm partitioning A, such that the restfiction fiB to each
cell BE V with B  A is continuous, and each cell in V is definable in the
model-theoretic structure (R, <, f(J)).
5. Let Xl," ., X k £; Rm be distinct nonempty definably connected sets and X their
union. Define a graph with vertex set {Xl,..., Xk} by putting an edge betw('l't:
Xi and X j (i =f j) if Xi n c1(X j ) =f 0 or cl(X i ) n XJ =f 0.

CELL DECOMPOSITION
59
ShoW that if XI(l)" .., XI(r) are the vertices of a connected component of this graph,
then XI(!) U ... U Xi(r) is a definably connected commponent of X, and that all
definably connected components of X are of this form. (This gives a useful method
to construct the definably connected components of X from a partition of X into
fi)litely many cells.)
Ii. Suppose g' is an o-minimal structure on (R, <) with g £; g', and let X £; R m
belong to g, so X also belongs to g'. Show that X is definably connected in the
oens e of g if and only if X is definably connected in the sense of g'.
7. Suppose g is an o-minimal structure on the ordered set (JR, <) of real numbers.
Show that for a definable set X £; JRm the following are equivalent:
(a) X is definably connected;
(b) X is connected in the usual topological sense.
REMARK. If g contains addition, then (a) and (b) above are also equivalent to
"X is definably path connected", see Chapter 6, (3.2), and also Chapter 6, (1.15),
('xercise 8.
tI. With the same hypothesis as in exercise 7, show that each definable set X C;;; JRm
is locally connected, that is, for each x E X and each open subset U of X containing
T there is a connected open subset V of X containing x and contained in U.
3. Definable families
(3.1) Let 5' C;;; Rm+n = Rm x R n be definable. For each aE Rm we put
Sa:= {x ERn : (a,x)ES}, a subset of R n .
We view S as describing the family of sets (Sa)a E Rm. Such a family is called a
definable family (of su bsets of R n , with parameter space R m ). The sets Sa are
also called the fibers of the family.
(3.2) EXAMPLE. Let n := (JR, <, +, .), and consider the formula
(*)
ax 2 + bxy + cy2 + dx + ey + f = o.
This defines a relation S C;;; JR8 xJR2. For each point (a,b,c,d,e,J)in JR8 the subset
S(a,b.o.d.e,]) of JR2 consists of the points (x, y) satisfying (*). We know that such a
s.et can be an ellipse, a parabola, a hyperbola, two intersecting lines, two parallel
lines, a single line, a single point, the empty set, or the entire plane JR2. In total
seVen different homeomorphism types occur among the sets in the family (Sa)aER'"
(3.3) In Chapter 9 we prove more generally for o-minimal expansions of ordered
rings that the sets of any given definable family belong to only finitely many de-
finable homeomorphism types. (Two definable sets belong to the same definable

60
CELL DECOMPOSITION
homeomorphism type if there is a definable homeomorphism between them.) This
is not true for arbitrary o-minimal structures: all intervals of any given o-minimal
structure belong to a single definable family, but given any cardinal" there are
o-minimal structures having" mutually non-homeomorphic intervals, see Chaptr
1, (6.6) and (7.11). Nevertheless, we will prove here some finiteness results 011
definable families in arbitrary o-minimal structures that are needed in the next
chapters.
(3.4) In the following 71': Rm+n -+ R m denotes the projection on the first m coordi-
nates.
(3.5) PROPOSITION.
(i) Let C be a cell in Rm+n and aE7I'(C). Then C a is a cell in R n .
(ii) Let V be a decomposition of Rm+n and a E Rm. Then the collection
Va := {C a : CEV, aE7I'(C)}
is a decomposition of Rn.
PROOF. For n = 1 this is immediate from the definitions. Suppose the propositioll
holds for a certain n, and let C be a cell in R m +(n+1). Let 71'1: R m +(n+1) -+ R m +"
be the obvious projection map, so that 71' 0 71'1 : Rm+(n+1) -+ Rm is the projection 011
the first m coordinates.
If C = f(1), then C a = f(fa), where fa: (7I'1C)a -R is defined by fa(X) = f(a,:r).
If C = (f,g)D with D = 71'1C, then C a = (fa,ga)E, where E = Da.
In both cases it is clear that C a is a cell in Rn+1. Property (ii) follows easily. U
(3.6) COROLLARY. Let S  Rm x R n be definable. Then there is a number Ms E 
such that for each a E Rm the set Sa  Rn has a partition into at most Ms cells.
In particular, each fiber Sa has at most Ms definably connected components.
PROOF. Take a decomposition V of R m + n partitioning S. Then for each a in }{I
the decomposition Va = {C a : GEV, aE7I'G} of R m consists of at most IVI cclb
and partitions Sa. So we can take Ms = IVI. 0
The following consequence is worth recording.
(3. T) COROLLARY. Let S  R m x R n be definable. Then there is a natural numhu
Ms such that fOT each a E R m the set Sa  R n has at most M s isolated points. II!
particular, each finite fiber Sa has cardinality at most Ms.
(3.8) The following exercise is intended for readers familiar with elementary logic.

CELL DECOMPOSITION
61
BXERCISE.
Let n = (R, <,...) be an o-minimal L-structure and R' = (R', <',...) an L-
structure elementarily equivalent to R. Show that R' is also o-minimal.
Notes and comments
The monotonicity theorem for o-minimal structures on the real line has an easy
proof, see Van den Dries [19]. The more general monotonicity theorem (1.2) is
due to Pillay and Steinhorn [49]. The cell decomposition theorem for "strongly"
o-minimal structures on the fealline is also in [19]. The general cell decomposi-
tion theorem (2.11) is again more difficult and was established by Knight, Pillay
and Steinhorn in [35]. It is worth keeping in mind that most proofs that a par-
ticular structure is o-minimal actually give at the same time the uniform finite-
ness property (2.13), that is, strong o-minimality. Thus we could have simplified
this chapter considerably for those readers who want to restrict their attention to
o-minimal structures on the real line for which the finiteness property (2.13) is
available (as is usually the case). However, in the model-theoretic literature on
o-rninimal structures it is important to have the results available in the present
generality, sometimes even in the sharper form given in (1.9), exercise 1, in (2.19),
exercise 4, and in the exercise of (3.8).

CHAPTER 4
DEFINABLE INVARIANTS: DIMENSION
AND EULER CHARACTERISTIC
As before we fix an o-minimal structure (R, <, S), and the usual terminological and
notational conventions of Chapters 1 and 3 remain in force. In this chapter we
establish the basic properties of the dimension and the Euler characteristic of a
definable set. Dimension and Euler characteristic are definable invariants for two
reasons; they are invariant under definable bijections, and they vary "definably"
in a definable family. At the end of Section 1 we apply results on dimension to
construct stratifications of definable sets.
1. Dimension
(1.1) We define the dimension of a nonempty definable set X  Rm by
dimX;= max{il+...+im: X contains an (i1,...,im)-cell}.
To the empty set we assign the dimension -00.
So dim X E {-oo, 0,1,..., m}, and dimX = m iff X contains an open cell. Par-
titioning X into finitely many cells we see that dim X = 0 iff X is finite and
nonempty. To prove that this dimension function has the right properties we need
the following.
(1.2) LEMMA. II A  Rm is an open cell and I; A..... Rm an injective definable
map, then I(A) contains an open cell.
PROOF. Clear for m = 1. Let m > 1 and assume inductively the lemma holds for
lower values of m. Taking a decomposition of Rm that partitions f(A) we have
f(A) = C 1 U ... U Ck for cells Cj in R m .
Then
A = r1(Cd u... u r1(Ck),
so at least one of the 1-1(C.), say f-l(Cd, contains a box B, and by taking B
suitably small we may assume that fiB is continuous. (By cell decomposition.) We
63

64 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
now claim that G 1 is open. If not, then by composing liB: B-+ C 1 with a definablp
homeomorphism of G 1 with a cell in R m - 1 we obtain a definable continuous injectiw.
map g:B-+Rm-1. Write B = B ' x (a, b).
Take c with a < c < b and consider the map h :B'_Rm-1 given by hex) = g(x, c).
By the inductive assumption applied to h we get h(B') 2 D for some box D il.
Rm-1. Let y be a point in D and take x in B ' with hex) = y.
If c' f:. c is sufficiently close to c, then g(x, c') will be in D, so g(x, c') = hex') '"
g( x', c) for some x' in B'. This contradicts the injectivity of g. 0
(1.3) PROPOSITION.
(i) If X  Y  R m and X, Y are definable, then dim X :$ dim Y :$ m.
(ii) II X  R m and Y  R n are definable and there is a definable bijection
between X and Y, then dim X = dim Y.
(iii) If X, Y  R m are definable, then dim(X U Y) = max{dimX, dim Y}.
PROOF. Property (i) is obvious. To prove (ii), let I:X - Y be a definable bijection
and d = dim X, e = dim Y. It is enough to show d:$ e since the reverse inequality
then follows by using f-1. Let A be an (i1"'" im)-cell contained in X, with
d = i 1 + '" + i m . Then 10 (PAl) : peA) -+ Y is an injective map and peA) an
open cell. Replacing X by peA), Y by I(A) and I by I 0 (PAl) we may as w('1!
assume that d = m and that X is an open cell in Rd. Let Y = C 1 U .., U Ck b..
a partition of Y = f(X) into cells. Then X = j-1(Cd U... U 1- 1 (Ck), so by tlH'
cell decomposition theorem 1- 1 (C;) contains an open cell B, for some i. Fix sucli
i and B.
Let C i = C £; R n be a (j1,..., jn)-cell. We shall prove that d :$ j1 + . .. + j".
(Since h +... + jn :$ e this will finish the proof.)
Suppose d > j1 + . . . + jn. The composition
B  C  peG) £; Rj,+"'+j.
is an injective map. Identifying Rj.+"+j. with a non-open cell (Rj,+...+j.) x {p}
in R d , where pERd-(j,+..+j.), we obtain a contradiction with lemma (1.2).
To prove (iii), let d = dim(X U Y), and let A be an (i 1 ,..., im)-cell contained in
X UY, with d = i1 +...+ i m . The open cell pA  R d is the union of PA(A n X) and
PA(A n Y), so by the cell decomposition theorem, one of these sets, say PA(A n X).
contains a box B in Rd. Then PAl (B) is an (it. . . ., i m )-cell contained in X, so that
dim X  d  dim X, i.e., dimX = dim(X U Y). 0
(1.4) Our definition of dimension was admittedly ad hoc, but is vindicated by its
invariance under definable bijections (property (ii) of the proposition above). As a

DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC 65
special case of this invariance, an (i 1 ,..., i", )-cell A hs diension i 1 + . . . + i",.
(Use the bijection PA between A and an open cell in R"+"'+'''.)
The next result says among other things that the dimension of a set from a definable
family depends "definably" on its parameters.
(1.5) PROPOSITJOl'i. Let S  R m x Rn be definable. For dE {-oo, 0,1,. .., n} put
S(d):= {aER"': dim Sa = d}.
Then S( d) is definable and the part of S above S( d) has dimension given by
dim ( U {a} x Sa ) = dim (S(d») + d.
aES(d)
PROOF. Let V be a decomposition of R",+n partitioning S. Let C be a cell in
V, with projection 'lrC £; R m . If C is an (i1,. . ., i m , . . ., im+n)-cell, then 'lrC is an
(i1"'" i",)-cell and C a is an (im+!,.. ., i",+n)-cell for each a E 'lrC. (By induction
on n, see Chapter 3, (3.5)(i) and its proof.) Hence,
(+) dimC = dim('lrC) + dim C a , for each a E 'lrC.
Consider now a cell A E 'lrV = {'lrC: C E V}, and let C1,..., Ck be the cells in V
that are contained in S and that project onto A, Le., 'lrC 1 = ... = 'lrCk = A.
For each a EA we have Sa = (C1)a u... u (Ck)a, so
dim(Sa) sup dim(C;)a
15i.k
sup (dimC; - dim A), by (*).
1:5;9
Calling this supremum d we note that dim Sa = d is independent of a E A, therefore
A £; S(d). So S(d) is a union of cells in 'Ir(V), hence S(d) is definable. ::-Jote also
that
d = ( SUP dim Ci ) - dim A = dim ( U C; ) - dim A
15;9 15i5 k
= dim ({a} x Sa) - dim A,
Le., dim (U aEA {a} X Sa) = dim A + d, and taking the union over all A E 'lrV with
A £; S( d) we get
dim ( U {a} x Sa ) = dim (S(d») + d. 0
aES(d)

66 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
(1.6) COROLLARY.
(i) dim S = maxo<d<n(dim S(d)+ d)  dim 1fS.
(ii) Let X  R n b definable and f : X.... R m a definable map. Then for each
dE{O,...,n} the set SI(d):= {aERm: dimf-1(a) = d} is definable and
dimf-1 (SI(d)) = dim SI(d) + d. Moreover, dim X  dim f(X).
(iii) dim(A x B) = dim A + dimB, for definable scts A and B.
PROOF. The equality in (i) is immediate from (1.5) and (1.3)(iii), and the inequality
from 1fS = UOdn Sed) and (1.3)(iii).
Property (ii) follows from (i) and, (1.5) by letting S = {(J(x),x): XEX}, taking
into account (1.3)(ii) and the fact that x...... (f(x), x) is a definable bijection betwecI:
X and S. Property (iii) is a special case of (i) by letting S = A x B. 0
::-Jote that in the situation of (ii) we have dim X = dim f(X) if each fiber f-1(a) is
finite.
We now turn to a more delicate result on dimension. Recall that for any definablp
set S £; R m we call cl(S) - S the frontier of S, to be distinguished from the
boundary bd(S) := deS) - int(S). Notation: as := deS) - S. We shall provc
below that dim as < dim S if S f:. 0. First a technical lemma.
(1.7) LEMMA. Let m > 0 and A  R m be definable. Then the set
A':= {x E R: a(A",) f:. (aA)x}
is finite.
PROOF. Note that a(A x )  (aA)", for all x E R. Assume for a contradiction that
A' is infinite. Then A' contains an interval I. After replacing A by A n (I x Rm-l)
we may assume that I = A' = 1f1(A), where 1f1 : R m --t R is the projection on the
first coordinate. For x E I, put F(x) := (aA)", - a(A",), so F(x) is a nonempty
subset of R m - 1 . Hence there exists a box B  Rm-1 such that B n F(x) f:. 0 and
B n Ax = 0. For any box B  Rm-1, put
1 B := {x E I: B n F(x) f:. 0 and B n Ax = 0}.
Let
G := {(a, b) = (a1,'.', a m -1 ,bl,... ,bm-I) E R 2 (m-1) : aj < bj for all i},
and let B(a, b) := (at, bI) x . ., x (am-I, b m -1) be the box in R m - 1 corresponding
to a point (a, b) E G. We now introduce the definable set
C'- U 1 B (a.b)x{(a,b)} (asubsetofR 2m - 1 ).
(a.b)EG
We shall establish two claims:
(1) The set IB(a,b) is finite for every (a,b) E G.
(2) int(C x ) f:. 0 for every x E I.

DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC 67
Claim 1 and (1.5) imply dim C ::; dim G = 2( m - 1), while claim 2 and (1.5) imply
dim C 2: 1 + 2( m - 1), a contradiction. Thus it only femains to pfove the claims.
(1) Suppose Claim 1 is false. Then there is a box B  R m - 1 and an interval J 
lB. By definition of IB we have (J x B) n A = 0, hence (J x B) n cI(A) = 0
since J x B is open. But for each x E J we have BnF( x) 'f: 0, so in particular
thefe exists y E B such that (x, y) E (J x B) n cI(A), a contfadiction.
(2) Let x E I. Then there exists a point (a,b) E CX' Thus F(x) n B(a,b) 'f: 0.
Take some y E F(x) n B(a,b). Then every point (a',b') E G such that
y E B(a',b')  B(a,b) also belongs to CX' Thus int(C x ) 'f: 0.
This finishes the pfoof of the lemma. 0
(1.8) THEOR.EM. Let S  R m be a nonempty definable set. Then
dim8S < dimS.
In particular dim (d( S)) = dim S .
PROOl'. By induction on m. The cases m = 0 and m = 1 afe obvious. So
let m > 1 and assume the result holds fOf lower values of m. We also assume
that dim S > 0, since otherwise the result holds trivially. For each i = 1,..., m
we considef the definable bijection 4>i : Rm -+ Rm defined by (jiJi(Xl,..., x m ) =
(Xi, Xl,.. ., Xi-I, Xi+l,.. ., x m ). We apply the preceding lemma to 4>i(S) to get a
fmite set F i  R such that 8(iPi(S)x) = (84)i(S)),. for all x E R - Fi. Put
IIi := 1I'j-l(Fi) = R i - 1 X F i X Rm-i, where 1I'i : Rm -+ R is the projection on the
ith coordinate, and put H := n::l Hi. Then H = F 1 x... x Fm is fjllite. MOfeover
m
8S  HU (8S)- H) = Hu U(8S)- Hi.
i=1
Thus by (1.3) it is enough to show that dim«8S) - Hi) < dim S for each i. Fix
some index i. Then
(jiJi(8S) - Hi) = U {x} X 8(iPi(S)x)'
,.eR-F,
By the inductive hypothesis dim8(iP;(S)x) < dim(4)i(S)t for all x E R for which
<l>i(S)x 'f: 0. Thus by the last formula displayed, taking only the union ovef the
x E R - F i such that 4>;( S):C 'f: 0:
dim (8S) - Hi) = dim iPi(8S) - Hi) < dim (jiJi(S) = dim S.
o
(1.9) COROLLARY. Let Sand T be nonempty definable subsets of Rm with S  T
and dim S = dim T. Then S has nonempty interior intT(S) in T and
dim(S - intT(S)) < dim S.
PROOF. Let dT denote the closure in T. Then S - intT(S) = S n clT(T - S) =
clT(T - S) - (T - S), which eithef is empty Of has by the theofem above dimension
< dim(T - S) ::; dim S. 0

68 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
(1.10) COROLLARY. Let S  R m be definable. Then dim(bd(S)) < m, whf:r'
bd(S) := cI(S) - int(S) is the topological boundary of the set S in R m .
PROOF. Clear if dim S < m. If dim S = m one uses
bd(S) ::; (cI(S) - S) u (S - int(S»)
together with theorem (1.8), and corollary (1.9) with T ::; R m . 0
DIGRESSION: EXISTENCE OF STRATIFICATIONS. (This material is not used later ill
the book.)
(1.11) DEFINITION. A stratification 6 of a closed definable set S  Rm is a
partition of S into finitely many cells, called strata of 6, such that for each stratum
A E 6 we have; 8A is a union of (necessarily lower-dimensional) strata of 6.
(1.12) Each partition of a closed definable subset of R into finitely many cpJls is a
stratification, but there are obvious examples of decompositions of R 2 that are lIot
stratifications.
(1.13) PROPOSITION. Let A  R m be a closed definable set and AI"", Ak dejill-
able subsets of A. Then there is a stratification of A partitioning each of AI,', ., A,.
First two easy lemmas:
(1.14) LEMMA. Let C,D be celis in Rm, where C is an (il"'" im)-cell and D £; ('.
Then the following are equivalent:
(i) D is an (i l ,. ..,im)-cell;
(ii) dim C = dim D;
(iii) D is open in C.
PROOF. A straightforward induction on m, just using the definition of cell. 0
(1.15) LEMMA. If A  R m is definable and C  A is a cell with dimC = dim A,
then there are cells DI" . ., Dk  C that are open in A such that
dim(C - (D! U... U Dk») < dimC.
PROOF. Write intA C = D! U...U Dk U D kH U...U Dt, where DI,..., Dk are cells
of the same dimension as C, and Dk+!,' . ., D, are cells of Jower dimension. Thrll
by lemma (1.14) each D i with 1 :5; i:5; k is open in C, hence open in intA C, helle<'
open in A. Moreover, C - (D! U ... U D k )  (C - int A C) U Dk+1 U .,. U D, which
has by (1.9) dimension < dimC. 0

DEFINABLE INVARIANTS: DIMENSION AND Et:LER CHARACTERISTIC 69
(1.16) PR.OOF OF (1.13). By induction on dimA. If dim A = 0 then A is finite
and the desired result is obvious. Suppose dim A > 0 and that the proposition
holds for definable sets of lower dimension. Take a finite partition l' of A into cells
that also partitions each Ai. Let 1'0 be the collection of cells in l' of dimension
dim A. After refining l' we may assume by lemma (1.15) that the cells of 1'0 are
open in A. Then A - U 1'0 is a closed set of dimension < dim A containing {} D for
each D E 1'0 and E for each E E l' - 1'0, so by the in ductive hypothesis there is a
stratification 6' of A - U 1'0 that refines l' - 1'0 and partitions the sets {}D for
all DE 1'0' ow set 6 := 1'0 U 6'. We claim that 6 satisfies the requirements.
By construction 6 is a stratification of A, and because 6" partitions each cell of
'P - 'Po and l' partitions each Ai, 6 partitions each Ai. 0
(1.17) EXER.CISES.
1. Let A  Rm be definable and 0 ::; d ::; m. Show that dim A  d if and only if
there is ad-tuple i = (i(I),. ..,i(d)) with 1::; i(l) < ." < i(d)::; m such that the
projection map Pi: Rm -+ R d given by Pi(X1,.. .,xm) = (Xi(l)'" "Xi(d») has the
property that Pi(A) has nonempty interior in Rd.
2. Let A <;; Rm be a definable set and a E Rm. Show there is a number d E
{ - DC, 0, . . . , dim A} such that dime UnA) = d for all sufficiently small definable
neighborhoods U of a in Rm, that is, for all definable neighborhoods of a in Rm
that are contained in some fixed definable neighborhood of a in R m .
The number d defined by this property is called the local dimension of A at a,
notation dima(A). Kote that dima(A) = -00 iff acI(A).
3. Show that if A is a d-dimensional cell, then dima(A) = d for all aEcI(A).
4. Let A <;; Rm be a definable set and d E {O,. . ., dim A}. Show that the set
{a E Rm: dim.(A)  d} is a definable closed subset of cI(A). Show also that if
A =f 0, then dim({aEcI(A): dim.(A) < d}) < d.
2. Euler characteristic
(2.1) Besides "dimension" there is another, more subtle, definable invariant, the
Euler characteristic. As a simple example, observe that a finite partition of an
interval into cells consists necessarily of k points and k + 1 intervals, for some
k  0, and that the quantity k - (k + 1) = -1 is independent of the partition
considered.
More generally, we assign to each cell C of dimension d the integer
E(C) := (_l)d,

70 DEFI:-lABLE INVARIA:-lTS: DIMENSIO:-l AND EULER CHARACTERISTIC
and given a finite partition 'P of a definable set S  Rm into cells we put
Eop(S) := L E(C) = ko - k 1 +... + (-l)dkd +... + (-l)mkm'
CffP
where kd is the number of d-dimensional cells in 'P.
We now have the following basic result.
(2.2) PROPOSITIOI". If 'P' is a second finite partition of S into cells, then we halJr
Eop(S) = £op.(S).
(2.3) We shall prove this below, see (2.8). Accepting the proposition for the mo-
ment we may define the Euler characteristic £(S) of 8 to be the common value of
Eop( 8) for finite partitions 'P of S into cells. The second basic fact is the invariance
of the Euler characteristic under definable bijections:
(2.4) PROPOSITION. If f:S-+R R is an injective definable map, then
E(8) = £(I(S)).
(2.5) This will be established in (2.12) below. Before starting the proofs of thcoc
propositions we single out certain finite partitions of cells and call these decom-
positions, generalizing the notion of decomposition of Rm from Chapter 3, (2.10).
The definition is by induction:
(1) All finite partitions of a cell C  R = Rl into cells are decompositions of
C;
(2) For m  1 a decomposition of a cell C  Rm+l is a finite partition V of C
into cells such that 7!'(V) .- {7!' D: D E V} is a decomposition of the «.11
7!'C R m .
Note: (2) also holds for m = 0 by considering the unique partition of the trivial
cell R O as a decomposition. The following fact is mentioned for the sake of COIII-
pleteness and will not be used: the decompositions of a cell C  R m are exactly
the restrictions to C of the decompositions of R m that partition C. (Exercise)
(2.6) LEMMA. If V is a decomposition of a cell C, then
E'D(C) = E(C) (= (_I)dlmC).
PROOF. By induction. Clear if C  R = R1. Let C  Rm+1, m  1, and assume
inductively
(.)
£..('D)(7!'C) = E(7!'C).

DEFINABLE INVARIANTS: DIMENSION A:-lD EULER CHARACTERlSTIC 71
We also assume that C is an (ii, . .. , i m , 1 )-cell; the case that C is an (it, . . . , i m , 0)-
cell is similar and left to the reader.
Let BE 1I"(V), so the list of distinct cells of V that map onto B under 11" is
r(1d, . . ., fUd, (10, ft), . . ., (It, 1t+1)'
for certain continuous definable functions /; on B. Let dim B = d. The contribution
of this list of cells to E'D( C) equals t. (_I)d + (t+ 1). (_1)"+1 = (_l)d+1 = -E(B).
Summinp; over the various BE 1I"(V) we get, using (*),
E'D(C) == - L E(B) = -E"('D)(1I"C) = -E(1I"C) == E(C). 0
Be>:-( 'D)
(2.7) LEMA. Every two finite partitions P(I) and P(2) of a definable set 5 s:;; R m
into cells have a common refinement to a finite partition P of 5 into cells such that
for each cell C E P( 1) U P(2) the restriction PIC is a decomposition of C.
PROOF. Take a decomposition of Rm that partitions each cell of P(l) U P(2).
Restricting this decomposition to 5 gives a partition P as required. 0
(2.8) PROOF OF PROPOSITION (2.2). Let P(l) and P(2) be finite partitions ofthe
definable set 5 s:;; Rm into cells. We have to show that Eop(I)(5) = Eop(2)(5). Take
a common refinement P of P(I) and P(2) with the property of the lemma above.
Then
Eop(5) == L Eoplc(C)
CEP(I)
= L E(C) by lemma (2.6)
CEP(I)
Eop(I)( 5),
1\nd in the same way we get Eop(5) = Eop(2)(5). 0
(2.9) As we noted in (2.3) we may now speak ofthe Euler characteristic E(5) ofa
definable set 5 without specifying a partition of 5 into cells. Clearly, for definable
sets 8 1 ,5 2 s:;; Rm we have
E(51 U 52) = E(5d + E(5 2 ) if 51 and 52 are disjoint;
i a general
E(5 1 U 52) = E(5d + E(5 2 ) - E(51 n 52),
the second formula following from the first by representing 51 U 52 as the disjoint
union of 51 - (51 n 5 2 ),51 n 52 and 52 - (51 n 52)'
N" ext we show that the Euler characteristic varies "definably" in a definable family
of sets.

72 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
(2.10) PROPOSITION. Let 5  Rm+" be definable. Then E(S..) takes only finitely
many values as a runs through the parameter space R m , and for each integer e thr.
set {a E Rm: £( 5a) = e} is definable. More precisely, let V be a decomposition
of Rm+" partitioning S and let 1f : Rm+" -+ R m be the projection on the first 111
coordinates. Given a cell A E 1f(V) there is a constant eA E Z such that £(Sa) = eA
for all a E Ai also
E(1f-I(A) n S) = E(A)eA'
PROOF. Suppose A E 1f(V) is an (i(I),. .., i( m»)-cell. Let C EV be a cell contained
in S with 1fC :::; A, so C is an (i(I),..., i( m), i( m+ 1),..., i(m + n»)-cell for certain
i(m+ 1),..., i(m+n). For each a E A the fiber C.. is an (i(m+ 1),..., i(m+n»-celJ,
so that £(C) = £(A). £(C..) for all a E A. Since 1f-I(A) n 5 is the union of celJs
C E V, and 5.. for a E A is the union of the corresponding cells C.., we obtain tile
constancy of £(5..) as well as the formula for £(1f- I (A) n 5). 0
(2.11) COROLLARY. Let S  Rm+" be definable and suppose all nonempty fibo'"
S.. (a E R m ) have the same Euler characteristic e and let 1f : R m +" -+ R m be as
above. Then
£(5) :::; £(1f5). e,
in particular E(A X B) = E(A). £(B) for definable A  R m and B  R".
(2.12) We can now start the proof of proposition (2.4). When finished we will have
removed the blemish that our Euler characteristic seems to depend on the notion
of "cell" which is not invariant under coordinate permutations.
Let 5  R m be definable and f: 5 -+ R" an injective definable map. We have to
show that £(5) = £(1(S»). (Note; we do not require continuity of /.)
By applying (2.11) to f(1) we see that £(S) = £(r(1». Let fl(f)::::; {(/(x),x) :
xE S} be the "reversed" graph of f. Again by (2.11) we have £(1(5») = £(f'(J»).
If we knew that £ (r(f») = £(f'(/») then we could conclude that £(S) :::; £(1(5)).
So we have reduced to proving the following:
Let U be a permutation of{I,...,m}; setting xu:= (X<1(l),...,X<1(m)) for x 
(x!,..., x m ) E R m , and Au := {xu: x E A} for A <;;; R m , we have £(A) = £(,.111')
for definable A <;;; R m .
Since the symmetric group on {I,.. ., m} is generated by the transpositions (i, i+ J)
with 1  i < m and A is a finite disjoint union of cells it suffices to prove this when
u is a transposition (i,i + 1) and A is a cell. For such U and A this will follow
if we can show that A is a finite disjoint union of subcells A :::; Al U '" U A<
such that (Adu,..., (Ak)U are also cells, because then dim Aj = dim Aju giv(',;
£(Aj) = E(Aju), so that £(A) = E£(Aj) = E£(Aju) = £(Au). So we are
done once the following proposition is established.

DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC 73
(2.13) PROPOSITION. Let C  R m be a cell and (f = (i,i + 1) a transposition,
1  i < m. Then C can be partitioned into cells Ct, . . ., Ck such that C 1 u, . . ., CkU
are also cells.
PROOF. By induction on m and dim C. Keep in mind that u = u- 1 .
Easy case: i < m - 1. Let us denote the restriction of u to {I,. . ., m - I} also by
(1, and write elements of R m as (x, y) with x = (xt,..., xm_dERm-1.
Subcase 1. C = (a,13)B. Then
(x,y)ECu # (xu,Y)EC
# xUEB and a(xu) < y < 13(xu)
# xEBu and au(x) < y < 13u(x),
where au:Bu-+R oo is defined by au(x) = a(xu), and similarly 13u.
Assuming inductively that the cell B is the disjoint union of cells Bt,..., Bk such
that B 1 U,. .., B/cu are also cells we see that Cu is the disjoint union of the k cells
(auIBlu,,8uIB1U), which equals (aIBl>13IBdu,
(au!B/cu,,8uIB/cu), which equals (aIB/c,13IB/c)u.
Subcase 2. C = 1'(13) with 13: B -+ R. The argument is similar.
Hard case: i = m - 1. Let C be a Ut, . . ., jm)-cell and B = 'lrC, A = 'Ir' B where
71' : Rm -+ Rm-l and 71" : R m - 1 -+ Rm-2 are the obvious projection maps. So B is
a Ul, . . ., jm-1 )-cell. We distinguish four subcases according to the values of jm-l
<tndjm. We write elements of R m as (x,y,z) with xER m - 2 .
Subcase 3. jm-l = jm = O. Then one checks easily that
C = {(x,j(x),g(x»): xEA}
for continuous definable functions j, g; A -+ R. Then
Cu = {(x,g(x),j(x»): XEA}
is dearly also a cell.
Subcase 4. jm-1 = O,jm = 1. Then B = r( a) for continuous definable a: A -+ R,
and C = (f, g) for continuous definable j, g; B -+ Roo. Hence
(x,y,z)EC # xEA and y=a(x) and j(x,y) < z<g(x,y)
# xEA and j(x,a(x» < z < g(x,a(x») and y = a(x).

74 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
Define J',g':A.....R"" by f'(x):;;;; I(x,a(x)) and g'(x) :;;;;g(x,a(x)), and define
a': (I', g')..... R by a'(x, z) :;;;; a( x). Then one verifies immediately that Cu = f(a' J.
so C u is a cell.
For the remaining cases we need some further notation and terminology: givplI
a definable continuous function I: (a,b)..... R on an interval we call c E (a,b) it
critical point of I if there are a "left" interval (I, c)  (a, b) and a "right" interval
(c,r)  (a,b) such that
either I is strictly increasing on (1, c) and I is decreasing on (c, r),
or I is constant on (l,c) and I is strictly monotone on (c,r),
or I is strictly decreasing on (I, c) and I is increasing on (c, r).
By the monotonicity theorem I has only finitely many critical points. Let tho,e
critical points be at < .., < ak and put ao = a, ak+! ;;;; b. Then we associate to f
the tuples c(f):;;;; (at,...,al:)ERI: and E(f):;;;; (EO,...,EI:) with EjE{-I,O,I} atlll
Ej;;;; -1 if I is strictly decreasing on (aj,aj+t),
Ej;;;; 0 if I is constant on (aj,aj+tJ,
Ej = +1 if I is strictly increasing on (aj,aj+tJ.
Subcase 5. jm-t = 1, jm = O. Then B = (a, j3)A and C = f(f) for a continuous
definable function I : B ..... R. The argument that follows is easy to visualize for
m = 2 in which case A = R O . The case for arbitrary m is essentially a "parametric"
version of the argument for the case m ;;;; 2, x E A being the parameter. Fur
each x E A we put c(x) := c(J(x,-)) and E(X) :== E(J(X,-)), where of cours
I( x, - ): (a( x), j3( x)) ..... R.
By partitioning A into finitely many subcells we may assume without loss of gPH-
erality that c(x) is of constant length k for x e A, that c(x):= (a1(x),...,ak(x))
is continuous as an Rk-valued function of x E A and that (x) is constant on A.
Then B is the disjoint union of the cells (aj,aj+d (0 :$ j :$ k, with ao :::: a and
al:+1 = 13) and the cells f(aj) (1:$ j :$ k), and again there is no loss of generality in
assuming B is actually one ofthose cells; when B == reaj) we are back in subcase 3,
so we may assume B = (aj, aj+d and we rename a :== aj and 13 := aj+1' So either
I(x, -) is strictly decreasing on (a(x),j3(x)) for all x E A, or f(x, -) is constanl
on (a(x),j3(x)) for aI] xeA, or f(x,-) is strictly increasing on (a(x),j3(x)) for aU
xEA.
Assume that I(x,-) is strictly increasing on (Q(x),j3(x)) for all xeA. (The other
two possibilities are handled similarly.) Define functions i,s:A.....R"" by
i(x) := inf{J(x,y): YE (a(x),j3(x))},
sex) := sup{J(x,y): yE (a(x),j3(x))}.
After partitioning A further into cells we may assume i and s are continuous 011 ;\.
and either R-valued, or identically -00, or identically +00. In any case i < son ;\.

DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC 75
For each x E A and z E (i(x),s(x)) let g(x,z) be the unique y E (a(x),,8(x))
such that f(x,y) = z, so f(x,y) = z $} g(x,z) = y. Then 9 is a definable
function on (i,S)A. We claim that 9 is continuous, from which the desired result
follows since C(J = f(g). Let f(x,y) = z, so g(x,z) = y, and let Yl>Yz E R
be such that a( x) < Yl < Y < Yz < (3( x). We have to find a neighborhood of
the point (x, z) in (i, s) that is mapped into the interval (Ylt yz) by the function
g. Choose elements Zl,ZZ E R with f(x,ytJ < Zl < Z < Z2 < f(X,Y2), and a
neip;hborhood U of x in A so small that if x' E U, then a(x') < Yl < yz < ,8(x'),
and f(x',yt) < Zl < Z2 < f(x',yz). Then C; X (Zl,ZZ) is the desired neighborhood
of (x,z): Let (x',z') E U x (Zl,ZZ). Since f(x',-) is strictly increasing there is
y' E (Yl, yz) with f(x', y') = Z'j hence g(x', z') = y' E (Yl, yz).
Subcase 6. jm-l = jm = 1. Then C = (It,fz)B, B = (a,(3)A. There are four
sub-subcases depending on whether or not It = -00, and whether or not fz = +00.
To shorten the exposition we do only one of these, leaving the other three to the
reader. (The ideas involved are similar.) So we shall assume that It is R-valued
and fz = +00, and we put f :;::; It, so C = (f, +00 )B. As in subcase 5 we may
reduce to the situation that f(x, -) is either strictly decreasing for each x E A, or
constant for each x E A, or strictly increasing for each x E A.
Assuming the last and defining i, 8: A -+ Roo as in subcase 5, we shall further assume
(as in subcaBe 5) that i and s are continuous and R-valued. (By partitioning A
we may reduce to this situation, plus a few other cases that are handled similarly.)
Then we define 9 :(i, s) -+ R as in sub case 5: g(x, z) = Y $} f(x, y) = z, so that 9 is
continuous. Now define 8':B -+ R by setting s'(x, y) := s(x). Then C is the disjoint
union of the cells (f, 8')B, f( s') and (s', +OO)B and we have (1, S')B(J = (a' ,g)(;,.)
where a': (i, s) -+ R is given by a'( x, z) = a( x)j f( s')(J and (s', +00 )B(J are similarly
seen to be cells. This completes subcase 6, and the proofs of propositions (2.13)
and (2.4). 0
(2.14) REMARKS. Given definable sets A s:;; Rffl and B s:;; R n , we have shown that if
there is a definable bijection between them, then dim A = dim Band E(A) = E(B).
In Chapter 8 we shall prove the converse for o-minimal expansions of real closed
fields. Without such an extra assumption one cannot expect a converse, since
intervals are all of dimension 1 and Euler characteristic -1, but there are divisible
ordered abelian groups in which some intervals are countable and other intervals
are uncountable, see Chapter 1, (7.11).
Note that we cannot have a definable bijection between a definable set A s:;; R ffl
and a subset A - {a}, aE A, since their Euler characteristics differ. This fact gives
a. negative answer to the following model-theoretic question:
Are there a definable set A s:;; R n and a definable map a : A -+ A such that
(A, a) == (N, S), where S is the successor function on the set N of natural numbers?

76 DEFINABLE INVARIANTS: DIMENSION AND EULER CHARACTERISTIC
(2.15) DIGRESSION: EULER CHARACTERISTIC OF CONSTRUCTIBLE SETS. (Th(.
material in this subsection is not used further on.)
Let K be an algebraically closed field. Recall that a constructible set in Km is it
finite union of sets of the form
{XEK m : f1(x) = .., = fr(x) = 0, gl(X):J 0,.. .,g.(x):J O},
where Ii, gj E K[X l , . . ., X m ]. The constructible sets in Km for m = 0,1,2, . . . f01"l1l
a structure on K, in fact they are exactly the sets definable in the field K usilll!;
constants, by Chevalley's constructibility theorem, see Chapter 1, (2.1).
Suppose now that K is also of characteristic O.
Then K = R(i) for some real dosed subfield R and i 2 = -1. (See for examplp
Lang [37] or the next volume.) Then we may identify K with R 2 by letting a T hi
correspond to (a, b) E R 2 . Then J(m gets identified with R2m, a constructiblp s(':
5  K m becomes a semialgebraic set 5  R 2m , and a constructible map I: S -+ 1\-"
becomes a semialgebraic map I: S -+ R 2n j as a semialgebraic set, 5 has an Eul(',
characteristic £(S), with £(5) = £(I(S)) if f is injective.
Does £(5) depend on the choice of the real closed subfield R?
In fact, £(5) is independent of the choice of R, and this is because the integer-
valued function £ (for any choice of R) satisfies the following rules:
(1) £(A U B) = £(A) + £(B) for disjoint constructible A, B  Km,
(2) E(B) = e . £(A) if I: B -+ A is a constructible map between constructibiP
sets A  J{m and B  J{n, and eEN is such that I/-l(a)1 = e for all aC A,
(3) £(J{m) = 1 for all m.
Using known properties of constructible sets and maps one easily checks that th('fp
is at most one integer-valued function £ on the class of constructible sets satisfying
these three rules. Hence E(A) does not depend on R.
We mention in passing that for constructible 5  Km the dimension dim S of S
considered as a semialgebraic set in R 2m is twice the dimension ofthe Zariski closure
of 5 in J(m, the last dimension being taken in the sense of algebraic varieties. 'fbi,
shows that dim 5 does not depend on the choice of Reither.

DEFINABLE INVARIA:-ITS: DIMENSION AND EULER CHARACTERISTIC 77
Notes and comments
\1u c h of the dimension theory in Section 1 has also been developed from a more
:uodel-theoretic viewpoint by Pillay [48], using an analogue of "transcendence de-
gree" for o-minimal structures.
Section 2 on Euler characteristic was inspired by a discussion with J. Denef, who
showed me that the Euler characteristic of constructible sets over C is invariant
under constructible bijections, and by a discussion with S. Schanuel, who told me
similar results on semilinear sets and semilinear bijections; see also [51]. This
suggested the possibility of a definably invariant Euler characteristic for definable
sets in any o-minimal structure. Digression (2.15) relates to Denef's remarks.
Strzebonski [59] uses the "o-minimal Euler characteristic" to obtain an analogue of
Sylow theory for groups definable in o-minimal structures.
Khovanskii called my attention to the following papers where Euler characteristic is
viewed as a finitely additive measure in a topological setting, and integration with
respect to Euler characteristic is developed and applied:
O. Va. Viro, Some integral calculus based on Euler characteristic, Topology and
Geometry-Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer-Verlag,
Berlin, 1989, pp. 127-138.
A.V. Pukhlikov and A.G. Khovanskii, Finitely additive measures on virtual poly-
lopes, St. Petersburg Math. J. 4 (1993), 337-356.
(This information came too late to include in the references at the end of the book
or influence the treatment in Section 2.)

CHAPTER 5
THE V APNIK-CHERVONENKIS
PROPERTY IN O-MINIMAL STRUCTURES
Introduction
The main result in this chapter is as follows, see (3.15) below.
THEOREM. Let (R, <, 8) be an o-minimal structure and S S;; Rp+q a definable set.
Then there is a positive integer d = d(S) such that for all sufficiently large n EN
each n-element set F S;; Rq has at most n d subsets of the form S" n F with x E RP.
This exhibits in a purely combinatorial way that the variation among the sets Sx is
highly restricted as x ranges over the parameter space RP: the total number of sub-
sets of an n-element set is 2 n , and this grows much faster with n than the polynomial
function n d . The key step (taken in Section 3) is to show that a model-theoretic
structure 'R- = (R,...) has this combinatorial property, if it has the property for
q = 1 (Shelah). The case q = 1 is easily checked for o-minimal structures, see (2.7)
and (2.11). We also derive some other results of a rather general nature.
The combinatorial fact expressed by our theorem has an interpretation in terms of
probability; see the notes at the end of the chapter. The ma.terial in this chapter
will not be used later on, but it would be a pity to omit it. Moreover, I believe
there is ample room for further developments along the lines of this chapter.
l. A combinatorial dichotomy
(1.1) Let C be a collection of subsets of an infinite set X. Given F <; X we put
CnF := {GnF: GEC},
the set of intersections of sets in C with F. If A C F is of the form A = G n F for
some G EC we also say tha.t A is cut out from F- by a set in C. So IC n FI ::5 2 1F1 ,
where ISI denotes the cardinality of a set S. Now we define a function fe : N - N
that in some sense measures the complexity of the collection C:
fe(n) := max{IC n FI: F is an n-element subset of X}.
So 0::5 fe(n) ::5 2 n for all n. We have the following, perhaps surprising, dichotomy.
79

80 THE VAPNIK-CHERVONENKIS PROPERTY IN O-MINIMAL STRUCTURES
(1.2) THEOREM. Either fc(n) = 2 R for all n, or else there is dEN such that
fc (n) :$ n d for all sufficiently large n.
The first possibility means that for each n there is an n-element set F  X all
of whose subsets are cut out from F by sets in C, while in the second case for al!
sufficiently large finite sets F  X relatively few subsets of F are cut out frolll F
by sets in C. This theorem will be derived from the following combinatorial lemrll a:
(1.3) LEMMA. Let F be a finite set of size n and V  'P(F), a collection of sulm t.,
of F, and suppose that
IVI > L G),
.<.
where 0 :$ d:$ n. Then F has a subset E such that lEI = d and V n E = 'P(E).
(1.4) REMARKS.
1. The hypothesis is sharp since the collection of subsets of F of size < d has
cardinality equal to the indicated sum of binomial coefficients, and this particular
collection violates the conclusion of the lemma.
2. Let us write the indicated sum of binomial coefficients as Pd(n). Clearly tllPI'('
is a unique polynomial Pd(X) E Q[X] (of degree d - 1 if d  1, Po(X) = 0), whose
value at n is Pd(n) for n  d. ote that then Pd(X) is defined for arbitrary real x,
and the "Pascal triangle equality" gives
PH(x-l)+Pd(x-l)=Pd(X) (dI).
(1.5) PROOF OF THE COMBINATORIAL LEMMA. By induction on n. The desired
result trivially holds for d = 0 and d = n, so let 0 < d < n. Pick a point x E F and
let F':= F - {x}. Also put D':= D - {x} for DEV, and let V' := {D': DEV}.
Note that under the map D ...... Dr : V ..... V' a set S E V' has either exactly 0111'
preimage or exactly two preimagesj in the latter case these two preimages ar/' S
and S U {x}. So V' = V 1 U V 2 (a disjoint union) where VI contains those S E I J '
having one preimage in V and V 2 those with two preimages. If IV' I > Pd(n - I)
then by the inductive assumption applied to F' and V' there is a subset E of F' of
size d with V' n E = 'P(E), hence V n E = 'P(E) and we are done.
So assume IV'I :$ Pd(n - 1). But
IVI = IVd + 21V 2 1 = (IVII + IVzl) + IVzl
= IV'I + IVzl > Pd(n) = Pd(n -1)+ Pd_1(n -1),
hence IVzl > Pd-1 (n - 1), so again by the inductive assumption applied to F ' and
V 2 there is a set E  F' of size d - 1 such that V 2 n E = 1'(E). Since for each set
SEV z we have SEV and Su {x}EV this gives vn (EU {x}) = 'P(EU {x}). 0
We now prove theorem (1.2) in the following more precise form.

THE VAPNIK-CHERVO:-lENKIS PROPERTY IN O-MINIMAL STRUCTURES 81
(1. 6 ) THEOREM. Suppose d is a nonnegative integer with Ic(d) < 2 d . Then
fc(n)::; Pd(n) lor all n.
PROOI'. If n < d then Pd(n) :::; 2" and the desired inequality holds trivially. So
let n  d, and let F be an n-element subset of X. If IC n FI > Pd(n), then the
lemma would give us a set E  F of size d such that IC n EI = 2 d , contradicting the
assumption le(d) < 2 d . Hence IC n FI ::; Pd(n). Since F was arbitrary, the desired
result follows. 0
REMARK. The inequalities in the theorem are sharp, since the collection C of sub-
sets of X of size < d satisfies le(d) < 2 d , Ic(e) = 2" for e < d, and le(n):::; Pd(n)
for all n.
2. Vapnik-Chervonenkis classes and dependence
(2.1) Given an infinite set X, a collection C  P(X) is said to be a Vapnik-
Chervonenkis class (or VC-class) if Ic(d) < 2 d for some dEN, and then we
define its Vapnik-Chervonenkis index V(C) to be the least such d; if C is not
a VC-class we put V(C) :::; 00. By theorem (1.6), V(C) < 00 means that Ie is of
polynomial growth, while V(C):::; 00 means that Ic(n) :::; 2" for all n.
(2.2) Usually a collection C  P(X) will be indexed by elements of a parameter
space or index set, and in fact the set X and this index set play dual roles. To
bring out this duality, let us assume that infinite sets X and Yare given, and a
binary relation   X X Y. For xEX and yEY we put
z:::::; {yEY: (x,y)E}, y::::; {xEX: (x,y)E}.
Let q,Y:= {y: yEY}  P(X) and x::::; {z:: XEX}  P(Y). (So Y is the
parameter space for'" Y and X is the parameter space for'" x.)
Let E  F <; X. Then we have for y E Y the obvious equivalence
E:::;"'Y n F $} YE"':z: for all xEE and Yrf....z: for all xEF - E,
hence
(*) EE...YnF$} ( nz: ) n ( n (Y-:Z:) ) of0.
:z:EE z:EF-E
As E ranges over the subsets of a finite set F <; X, the nonempty intersections
on the right in (*) are exactly the atoms of the boolean algebra of subsets of Y
generated by the "'z: with x E F. So we conclude (for finite F  X):
{ I'" Y n FI = number of atoms of the boolean algebra of
subsets of Y generated by the sets "'z: with x E F.
(**)

82 THE VAPNIK.CHERVONE:-lKIS PROPERTY IN O-MINIMAL STRUCTURES
(2.3) This leads us to a situation dual to the earlier set-up: Fix a set Y. For
S  Y we write Sl :== S, and S-l :== Y - 5 (the complement of S in V). Giv<'n
sets Sl,', ., Sn  Y we note that the 2 n sets (many of which may be empty)
S:(l) n... n s(n), f:{1,.. .,n}-+{-I,l},
are disjoint and cover Y, so the boolean algebra B( Sl, . . ., Sn) of su bsets of Y I\('n.
erated by Sl,..., Sn has as its atoms the nonempty sets among these intersections.
Hence the maximal number of atoms is 2 n and this is achieved precisely when all
2 n intersections indicated are nonempty. In this case we say that the sequenq,
8 1 , . .., Sn is independent (in V), or (abusing language) that Sl,''', Sn are in.
dependent (in V). Otherwise we say that the sequence 51"", Sn is dependent.
We also say that a collection 9 of subsets of Y is independent (in Y) if tor ea('h
n E  there is an independent sequence 51,"" Sn in 9, and otherwise we call g
dependent. The following is now dual to theorem (1.6).
(2.4) PROPOSITION. Suppose 9  P(Y) is dependent, so there is d such that Wf'},
sequence Sl,', ., 8d in 9 is dependent. Then for such d and all Sl,. . ., Sn E 9, till
boolean algebra B( Sl, . . ., Sn) has at most Pd( n) atoms.
PROOF. Let n  d and 5 1 "",Sn E9. Let F:= {1,...,n}, and let 1)  P(I-')
consist of the sets D  F for which the intersection
( n 5 i ) n ( n 8i- l )
ieD ij!D
is nonempty. If 11)1 > Pd(n) then by lemma (1.3) there would be a set E  {1,.. '. 1/)
of size d such that P(E) = 1) n E, hence the boolean algebra generated by the s('I,
Si with i E E would have 2 d atoms, contradicting the assumption on d. 0
(2.5) The simplest examples of dependent collections are obtained as follows. \,('1
B be a collection of subsets of Y and d a positive integer such that each nonempl,1'
intersection B l n. . . n Bd of d sets from B equals an intersection n i e D B j for 50111<'
proper subset D of the index set {I,.. .,d}. For instance, this is the case for Boo
the set of connected subsets of the real line R, with d = 3. More generally, given a:l
o-minimal structure (R, <, S), we take Y = Rand B = the collection of defilla!>ly
connected subsets of R; then B satisfies the property above with d = 3, as is eaoily
checked.
(2.6) LEMMA. Let Band d be as in the general statement of (2.5). Let 9  'P( \')
and suppose e E N is ,quch that each set in 9 is a boolean combination of at most (
sets in B. Then 9 is dependent. More precisely, B( 51, . . ., Sn) has at most Pd (I f/)
atoms, for all 51, . . . , Sn E 9.
PROOF. Let each Si be a boolean combination of sets Bil,...,B.. in B, and kl
F := {( i,j); 1 ::; i ::; n, 1 ::; j ::; e}, an index set of en elements. Then the booJc<ln

THE VAPNIK-CHERVONENKIS PROPERTY I:-l O-MINIMAL STRUCT1:RES 83
algebra generated by the Si'S is contained in the boolean algebra generated by the
I/).'s with>' E F, and the latter boolean algebra has as atoms the nonempty sets
among the intersections
C D B).) n ( n {B;l : ilE F, BjJ does not contain). 0D BA})
where D varies over the subsets of F of size < d. Hence there are at most Pd(en)
atoms in the boolean algebra generated by the B). 's with>' E F, and hence at most
P.( en) atoms in the boolean algebra generated by Sl, . . . , S n' 0
(2.7) EXAMPLE. Let (R, <,8) be an o-minimal structure and if>  RmH a defin-
able set and put Q :== {if>z:: x E Rm}, a collection of subsets of R. By Chapter
3, (3.6) we know that for some fixed e EN each set in Q has at most e definably
connected components. Hence by taking for B the collection of definably connected
subsets of R the lemma shows that Q is dependent, and that the boolean algebra
cnerated by sets Sl, . .., Sn E Q has at most P3( en) atoms. Later we shall see that
for each definable set if>  Rm+n the collection if> x := {if>z:: x E Rm} of subsets of
R n is dependent.
(2.8) Given an infinite set Y and a collection Q  P(Y) we define a "growth
function" fl1: N -+ N by
fl1(n) := maximum over all Sl"'" Sn E Q of the
number of atoms of B(Sl,"', Sn)'
In the case where Q is dependent, we let D(Q) be the smallest dEN for which
fQ(d) < 2 d , and we call D(Q) the dependency index of Q. If Q is independent
we set D(Q) = 00.
(2.9) Suppose X and Yare infinite sets and if>  X x Y is a binary relation.
Then we defined in (2.2) above the collections C = if>Y  P(X) and Q = if>x 
P(Y), and (+) and (++) show that fc = fl1, hence V( iji Y) = D( iji x), and in
particular,
if>Y is a VC-class $} iji x is dependent,
and by reversing the roles of C and 9,
iji Y is dependent $} if> x is a VC-class.
Less obvious perhaps is
(2.10) PROPOSITION. if>Y is independent if and only if if> x is independent.
PROOF. By symmetry we need only prove one direction, so assume if>Y is indepen-
dent, and let n E N. We want to find x(I),.. ., x(n) E X such that ijiZ:(l)"'" if>x(n) 

84 THE VAPNIK-CHERVONENKIS PROPERTY IN O-MINIMAL STRUCTLRES
Yare independent. We know there are 2 n independent sets y(.) (e E {-I, l}n) in
X, so in particular there is for each mE {I, .. . , n} a point x( m) in X with
x(m) E (.(0=1 y«)) n CmQ-l (y«)rl) ,
that is, (x(m),y(e»E if> <* e(rn) = 1, for all eE{I, _1}n, hence
y(e)Ez:(m) <* f(m) = 1,
and thus
y(e) E n (r(m)r(m).
l:Sm:Sn
Therefore if>Z:(I)"'" if>z:(n) are independent. 0
(2.11) Because of this proposition and the equivalences in (2.9) the properties "4>1'
is a VC-class", "if> Y is dependent", "if> x is a VC-class", "if> x is dependent" are all
equivalent. The proof of (2.10) shows moreover
D(if>Y»2 n :::} D(if>x»n,
hence
D(x)  n :::} D(Y)  2 n ,
or in terms of C = if> Y alone,
D(C)  2 V (C).
We call the binary relation  dependent or independent, according to wheth,'r
if>x is dependent or independent, and we set D(if»:= D(if>x) = V(Y).
A useful source of VC-classes is the following result from Dudley [24].
(2.12) PROPOSITION. Let £ be an m-dirnensi01lal real vector space of real-valll(.d
functions on an infinite set X, and for each f E l" put pose/) := {x EX:
f(x) > OJ. Then pos(l,) := {pos(f): f E l,} i.q a VG-class of subsets of X
with V(pos(l,») = m + 1.
PROOF. Let A  X be a set with IAI = m + 1. We shall see that then not cae:l
subset of A is cut out from A by a set in pos(£). The restriction map f ,.... flA: L 
R A cannot be surjective since dim(R A ) = m+ 1 > dim(£). Therefore R A containo a
nonzero vector w that is orthogonal to all flA (fEl,) with respect to the standard
inner product (u,v) := LaEA u(a). v(a) on IRA. Replacing w by -w if necessary
we may as well assume that A+ := {a E A: w(a) > O} is non empty. If there wert'
fEl, with A+ = An pos(J), we would have 0 = (w,fiA) = Lw(a).f(a) > 0, a
contradiction. Hence A+ is not cut out from A by any set in pos(£). This shows

THE VAPNIK-CHERVONE:-lKIS PROPERTY IN O-MINIMAL STRUCTURES 86
v{pos(£») ::; m + 1, and we leave the proof that V(pos(£)) is not less than m + 1
as an exercise. 0
(2.13) In particular the proposition applies to the vector space of real polynomial
functions on X = ]RN of degree::; d, where N is a positive integer. In the exercises
below this leads to a quite elementary proof that every semialgebraic binary relation
4.>  ]R M X ]R N (M, N > 0) is dependent. (This is also a special case of the main
theorem ofthis chapter.)
(2.14) EXERCISES.
1. Finish the proof of (2.12) by showing that "Ii (pos(£»  m + 1.
Let X, Y be infinite sets. Given +  X x Y, put f' := fl1:N .....N, where 9 = + x,
so f'(n) = the maximum over all x(I),.. .,x(n) in X of the number of atoms of
the boolean algebra B(+x(l)"." +:r(n»)'
Let +, qs  X x Y, and define the relations "'+, + V '1', +&'1'  X x Y as follows:
(x, y) E"'+ iff (x, y)  +, (x, y) E + V 'I' iff (x,y) E + or (x, y) E '1', (x, y) E +&'1' iff
(x,Y)E+ and (x,Y)Eqr. Note that f' = r ' .
2. Show that f'v" :$ f' . f", and derive that fit... :$ fit . ffl .
3. Show that if + and 'I' are dependent, then + V 'I' and +&'1' are dependent.
4. Derive from the previous two exercises and proposition (2.12) that every semi-
algebraic relation +  ]RM X ]RN (M, N > 0) is dependent.
3. Reduction to the case q = 1
In this section we prove the following rather difficult result due to Shelah [53]:
(3.1) THEOREM. Let n = (R,...) be an infinite model-theoretic structure and
suppose all definable relations +  RP x R, for all p > 0, are dependent. Then all
definable relations +  RP X Rq, for all p, q > 0, are dependent.
(3.2) Following Laskowski [38J we establish this in a purely combinatorial way, via
a somewhat more precise result than (3.1), namely theorem (3.12) below. As an
essential tool we first prove a well-known combinatorial fact due to Ramsey.
(3.3) Given a set X and a positive integer r, we put
x(r):= the collection of all r-element subsets of X.

86 THE VAP:-lIK-CHERVONENKIS PROPERTY IN O-MI:-lIMAL STRUCTURES
(3.4) RAMSEY'S THEOREM. Given positive integers M, rand k there is a positiVe
integer N = N(M, r, k) so large that if X is a set with IXI  Nand x(r) ==
P l U Pz u... U P k , then there is a set Y  X with WI = M such that y(r)  Pj for
some jE{I,.. .,k}.
PROOF. Clearly it suffices to prove this for k = 2, and we do this cCUje by induction
on r. If r = 1 we can take N = 2M. Assume true for a certain r, so we have
positive integers N(M, r, 2) with the desired property. Next define positive integPfs
N(I),..., N(2M) by descending recursion:
N(2M) = 1 and N(i):= N(N(i + 1),r,2) + 1 for 1:5 i < 2M.
We show that then N = N(M, r + 1,2) := N(I) has the required property. Let
X = {I,..., N}, with N = N(I) as above, and let x(r+1) = P l U Pz. We construct
a descending sequence Al '2 A2 '2 .., '2 AZM of subsets of X, with IAil  N(i).
and a sequence al,...,aZM, with ai E Ai for i = 1,...,2M, and ai fj. A i +1 fOf
i < 2M. To start with, put Al = X = {I,. .., N}, and let al be any element of /11.
Suppose Ai and ai E Ai have been constructed, jAil  N(i), 1 :5 i < 2M. TIH'n
(Ai - {ai})(r) = Ql U Q2, with
Ql .- {{bi>" .,b r } E(A i - {ai} )(r): {ai, b l ,... ,b r } E P l },
Qz := {{bi>..., b r } E (Ai - {ai} )(r): {ai, b l ,... ,b r } E P2}'
Since IAi - {ai}i  N(N(i + 1),r,2), the inductive hypothesis implies there is
Ai+l  Ai - {ai} with IAHll  N(i + 1) and A\l  Ql or Al  Qz. Let Ui+1
be any element of Ai+l' This finishes the construction of the Ai'S and ai's. let
A = {al,..., a2M}, and note that for each iE {I,..., 2M},
either (1) {ai, ai(l),' .., ai(r)} E Pl whenever i < i(l) < ... < i(r) :5 2M,
or (2) {ai, ai(l),"', ai(r)} e P 2 whenever i < i(l) < .. . < i(r) :5 2M.
Let Yi := {ai: (1) holds} and Yz := {ai: (2) holds}. Then A = Yi U Y z , and
either 11'1 I  M or IYiI  M, and clearly Yl(r+l)  P l and Yz(r+l)  Pz. So eitlH,1'
Y = 1'1 or Y = Yz has the required property. 0
(3.5) Ramsey's theorem is intimately related to the model-theoretic notion of in-
discernibility, which we need here only in a very simple form, as follows.
DEFINITIOro;S. Let X be an infinite set. Given a relation A  X r we say that a
finite sequence Xl,' . ., X M in X is A-indiscernible if
(Xi(l),,,,,Xi(r»)eA $} (Xj(l),...,Xj(r») eA.
whenever 1 :5 i(l) < .., < i(r) :5 M and 1 :5 j(l) < .., < j(r) :5 M. Let A he it
collection of relations on X, that is, each element of A is a. set A  X r , with r E /II
depending on A. Then a sequence Xl, . . ., X M in X is called A-indiscernible if it
is A-indiscernible for each A EA. A subsequence of a sequence Xl,' .., XN is by
definition a sequence Xi(l),' .., Xi(M) with 1 :5 i(l) < . .. < i(M) :5 N.

THE VAPNIK-CHERVO:-lENKIS PROPERTY I:-l O-MINIMAL STRUCTURES 87
(3.6) COROLLARY. Let X be an infinite set and A a finite collection of relations
on X. Then, given any positive integer M there is a positive integer N so large that
each sequence in X of length N contains an A-indiscernible subsequence of length
M.
PROOF. By an induction on IAI it suffices to prove this when A consists of just
one relation A  xr, r > O. Let Xl,' . . , X N be a sequence in X oflength N, with
N == N(M, r, 2), and write {I,..., N}(r) == PI U P2 with
P I .- {{i(I),...,i(r)}: Ii(I)<...<i(r)N, (Xi(1),...,Xi(r»)EA},
P 2 .- {{i(I),...,i(r)}: 1  i(l) <... < i(r)  N, (Xi(l),,,,,Xi(r»)A}.
By Ramsey's theorem there is {i(I),..., i(M)}  {I,..., N}, with i(I) < ... <
i(M), such that {i(I),...,i(M)}")  PI or {i(I),...,i(M)}(r)  P 2 . Then the
subsequence Xi(l),' .., Xi(M) of Xl>.' ., XN is clearly A-indiscernible. 0
(3.7) .For the rest of this section we fix infinite sets X and Y and a binary relation
4.> XxY.
(3.8) LEMMA. Suppose if> is independent, and A is a finite collection of relations
on X. Then there are for each MEN an A-indiscernible sequence ai, . . ., aM in X
and an element bE Y such that for all mE {I,. . ., M}, (am, b) E if> $} m is even.
PROOF. Fix a natural number M. By (3.6) we can find a. natural number N so
l1\rge that each sequence in X of length N contains an A-indiscernible subsequence
of length M. As if> is independent there are elements x( i) E X for I  i  N such
that the sets if> x(I), . .. , if> x(N) are independent; so for each set w  {I,. . ., N} the
intersection
(1 )
( .n if>X(i» ) n ( .n (if>X(i») -1 )
'E w .j!w
is nonempty.
Fix an A-indiscernible subsequence x( id,..., x( iM) of length M of (X(i»)l<i<N'
and take for b an element of the intersection (1) for w := {i 2m : I  m  MT2}.
Then the lemma above holds for the sequence ai, . . . , aM where am := x( i m ). 0
(3.9) Next we introduce certain relations that are in some sense "definable" from
. Let the variables Xl,' . . , X M range over X and the variable y over Y.
Given a positive integer M and a set u  {I,..., M} we put
+u(X1,...,XM;y):= ( 1\ if>(Xi,Y» ) & ( 1\ ...if><Xi,Y» ) ,
1 E u 't!u

88 THE VAPNIK-CHERVONENKIS PROPERTY I:-l O-MINIMAL STRUCTURES
a formula defining a subset cli" of XM x Yj the image of cli" under the projection
map X M x Y ..... X M is defined by the formula 3ycli,,(ZI" ",ZM;Y), and here w
simply regard this last formula as a convenient notation for the subset of XM it
defines. So
A.,M:= {3ycli,,(ZI,...,ZMjY): u  {1,...,M}}
is simply a finite collection of M-ary relations on X.
The following is a converse to the previous lemma.
(3.10) LEMMA. Let a(l),.. ., a(N) be a sequence in X that is At.M-indiscerniblr.
N  2M, let a( iI), . . ., a( iZM) be a subsequence, and let bEY be such that for all
mE{I,...,2M},(a(im),b)Ecli $} miseven. ThenD(cli»M.
PROOF. It suffices to show that clia(l)'" .,clia(M) are independent. Note first that
cli E(a(iI),. .., a(izM)j b) holds for E := set of even integers in {l,.. ., 2M}. Giv"l1
any set u  {I,..., M}, take a sequence 1 $; k(l) < ... < k(M) $; 2M such tha:
k( i) is even for i E u and k( i) is odd for i ilu. Then clearly
cli,,(a(ik(l»'" .,a(ik(M»jb)
holds. Hence 3ycli,,(a(i k (I»),...,a(i k (M»jy) holds, and thus
3ycli" (a(l),..., a(M); y)
holds by indiscernability. Since u  {1,...,M} was arbitrary this shows tha:
clia(I)," ., clia(M) are independent. 0
(3.11) Next we assume also that the set Y is given as a cartesian product }'
YI X Yz, with both Y 1 and Yz infinite. Let the variable YI range over Y 1 and Yz <JV('[
Yz. Consider now the formula cli(z; YI, yz) as defining a relation clio  (X X Y 1 ) X }'2'
(This is just cli if we identify (X x YI) X Yz with X x (Y I x Yz) in the usual way.)
The relation clio parametrizes a collection of subsets of Yz with index set X x h,
and so it makes sense to say that clio is dependent.
Given a positive integer M and a set u  {I, . . ., M} we introduce a formula
r",,(XI'" .,ZMiYI) := 3Y2 cli ,,(ZI," .,XM;yI,YZ)'
This formula r.,,, defines a subset of XM x Y I , and we will simply regard r."
as a notation for this subset. So r.." parametrizes a collection of subsets o. Yt
with index set X M . So it makes sense to say that r.,,, is dependent. With tll<"('
notations we have
(3.12) THEOREM. Suppose there are positive integers M and N such that D(<I>"):S
M and D(r .,,,) $; N for all u  {l,. . ., M}. Then if> is dependent.
REMARK. The constructive nature of the proof below and of the arguments ahu vr
would in principle allow us to extract a (huge) bound D(if»:S B(M,N), for SOIll"

THE VAPNIK-CHERVONENKIS PROPERTY IN O-MI:-lIMAL STRUCTURES 89
eXplicit function B of M and N. The construction of this function B would involve
the bounds in Ramsey's theorem.
PROOF. We introduce a formula qru,v for u  {I,..., M} and II  {I,..., N}:
qru,V(Xl,...,XN):= 3Yl ( /\ r.,u(xj;Yd & /\ ...,r.,u(xi;Yd ) ,
jEv jf!v
w here each xi = (xi>,.. ., XiM) is a variable ranging over X M. SO 'II' u,V defines a
subset of (XM)N = X MN . Actually qru,v is just 3Yl(r.,u)v(x 1 ,..., XNj Yd in the
notation of (3.9) above. Put Au : {qr..,v: II  {I,.. .,N}} and
A:= {qru,v: u  {1,...,M}, II  {l,...,N}},
so A is the union of the Au's.
Suppose IJ? is independent. (We shall derive a contradiction.) Put
K := (2N)2 M .2M.
By lemma (3.8) there are an A-indiscernible sequence al,.", aK in X and an
element b = (b 1 ,b 2 ) E Y 1 X Y2 such that lJ?(ak;b) holds if and only if k is even.
Since D( IJ?) $; M there is by lemma (3.10) no interval J of length 2M contained
in {I,. . ., K} 80 that the sequence (Uk, btJk E J is A.. ,M-indiscernible.
Each relation in A..,M is defined by a formula of the form
3y 2 1J?: (Xl, Yl1),"', (XM, Y1M)i Y2)
with u  {I,. .., M} and each variable (Xi, Yli) ranging over XxY 1 . For (Uk, b1h E J
to be indiscernible with respect to this relation is equivalent to (akh E J being
r',..(Xl,' .., XM; bt}-indiscernible. (Here r.,..(Xl"'" XM; btJ stands for the subset
of X M defined by it.) So there is no interval J of length 2M contained in {I, . . ., k}
such that (ak)k E J is r.,u(Xl"'" X m ; btJ-indiscernible for all u  {I,..., M}.
CLAIM. Let P and Q be positive integers with Q  2N P, let 1= {k : io $; k <
io + Q} be an interval of length Q contained in {I,..., K}, and let u  {I,. . ., M}.
Then I has a subinterval J := {k: jo $; k < jo + P} of length P, such that the
sequence (ak)/oE J is r..u(Xl,"" XMi btJ-indiscernible.
Note that if the claim holds, then starting with Q := K and P := Q /2N, and
successively applying the claim to each of the 2 M subsets u  {I,.. ., M} we obtain
!l the end an interval J of length 2M contained in {I, . . ., K} so that the sequence
,Qk)k E J is r..u(Xl,..., XMi bt}-indiscernible for all u  {I,..., M}, and we have
a Contradiction as desired. So all that remains is to prove the claim.

90 THE VAPNIK-CHERVONENK1S PROPERTY 1:-< O-MINIMAL STRUCT1:RES
PROOF OF CLAIM. Assume there is no such interval J. Now, given any j with
0:5 j < 2N the interval J(j):= {k: io+ jP:5 k < io+U+l)P} is oOength P and
contained in I, so in particular (akh E J(j) is not r.,,,(ZI,.", ZM; bd-indiscerniul..
So for some strictly increasing sequence of length M in J(j) the corresponding
sequence of a's does not satisfy r.,,,(ZI,...,XM;bd. When j is even we choo""
a strictly increasing sequence k(j,I) < ... < k(j, M) in J(j) of the first kind,
while for odd j we choose k(j, 1) < '" < k(j, M) in J(j) of the second kind.
Hence r.,,, (ak(j,l),'''' ak(j,M); b 1 ) holds if and only if j is even, for 0 :5 j < 'IN.
Put aj := (ak(j,1) , ..., ak(j.M)) E X M for 0 :5 j < 2N. Since (ak)kE 1 is all A
indiscernible sequence, the order in which the ak's appear within an "increasinf!;"
sequence of aj's implies that (aj)0:5j<2N is Au-indiscernible. But r.,,, (aj; bd hold,
if and only if j is even, which by lemma (3.10) contradicts D(r..,,) :5 N. D
(3.13) We call attention to the fact that for dependence of. we need only dep('r:-
dence of the auxiliary relations .. and r.. u , which are defined purely in terms of
.; also, the definitions of.. and r.,,, in terms of. do not involve quantifiers uv('r
X.
(3.14) PROOF OF THEOREM (3.1). We are given an infinite model-theoretic strllc
ture n = (R,...) such that each definable rela.tion . <;;; RP x R, for each p > 0,
is dependent. We have to show that all definable relations. <;;; RP x Rq arc (1..-
pendent, for all p, q > O. The case q = 1 holds by assumption. Assume inducti\'f'ly
that the desired property holds for a certain q > 0, and let. <;;; RP X Rq+l u..
definable, p> o. Now set X :== RP, Yt := Rq, and Y2 := R, Y := 1'1 X Y 2 = if}':.
Then. <;;; X x Y, so we are in the setting of (3.11) above. By the hypothesis of
the theorem the (definable) relation.. <;;; Rp+q x R is dependent, say D(..)  ,\1.
for a certain positive integer M. Since clearly all relations r.." <;;; RMp X Rq wit.h
u <;;; {I,. . ., M} are definable, the inductive assumption implies there is some posi
tive integer N such that D(r.,,,) :5 N for all u <;;; {I,..., M}. Then theorem (3.1'2)
tells us that. is dependent, as desired. D
(3.15) COROLLARY. Let (R, <,8) be an o-minimal structure and if> <;;; RP X 11 1
(p, q > 0) a definable relation. Then. is dependent. In particular, there i d --"
d(.) E N such that for all sufficiently large 11 each n-element set F <;;; Rq has at
most n d subsets of the form if> x n F with x E RP.
PROOF. From (2.7) it follows that (R, <,8) satisfies the hypothesis of thco/(,II J
(3.1). Hence. is dependent. Thus if> X == {.x: x E RP} is a VC-class, by
(2.11). 0

THE VAPNIK-CHERVONENKIS PROPERTY IN O-MINIMAL STRUCTURES 91
Notes and comments
The combinatorial dichotomy of Section 1 is due to Shelah [53], and independently
to Vapnik and Chervonenkis [62]. In [53] this had to do with the model-theoretic
"independence property", and in this connection Shelah also proved (3.1) by a
curious set-theoretic argument.
In [62] the motivation came from probability theory: if C is a VC-class of events in
a. probability space X, then the estimates in Bernoulli's theorem for the events in
C are uniform in a certain technical sense. See also Dudley [24] for more on this.
That o-minimal structures do not have the independence property was shown by
Pillay and Steinhorn [49]. The combinatorial interpretation in terms of the VC-
property was provided by Laskowski [38] who also gave the purely combinatorial
proof of (3.1) in Section 3. Wilkie (unpublished) gave another proof of (3.1) for
o-minimal structures using their special properties. Somewhat earlier Stengle and
Yukich [58] had shown (as in (2.13) and (2.14) above) that semialgebraic collections
have the VC-property.
For recent applications to the study of neural networks, see Macintyre and Sontag
[41] and Sontag [56].

CHAPTER 6
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
Introduction
In this chapter we fix an o-minimal expansion (R, <,8) of an ordered abelian group
(R, <,0, -, +). (As was shown in Chapter 1, this group operation makes R an
abelian divisible torsion-free group, so we may and do consider R as a vector space
over IQI.)
For convenience we fix a positive element 1 E R. We also set
{ X if x  0,
Ixl:= -x if x < O.
The presence of such a group operation leads in Section 1 to quick proofs of curve
selection and of the fact that a definable continuous map on a closed bounded
definable set ha.s closed bounded image.
Definable curves here playa role similar to that of sequences in R n, but have better
properties. Section 2 proves that "fiberwise open" implies "piecewise open" for
definable sets S  Rm+n, and variants of this important fact. In Section 3 we show
that "definably connected" equals "definably path connected", and we obtain some
results on definable partitions of unity. In Section 4 we consider definably proper
and definably identifying maps.
1. Curve selection
(1.1) An important consequence of the presence of the group structure is that we
can definably pick an element e(X) E X from each nonernpty definable set X. The
idea is to let e(X) be the midpoint of X if X is a bounded interval. Here are the
details:
(i) Let X  R be definable and nonempty. If X has a least element, then we
let e(X) be this least element. If X does not have a least element, let (a, b)
be its "left-most" interval: a = inf X, b = sup{ x E R: (a, x)  R}. Then
93

94
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
a < b and (a, b)  X; now put
{ o
b - 1
e(X);== a + 1
(a+b)/2
if a == -oo,b == +00,
if a == -00, bER,
ifaER, b==+oo,
if a,bER.
(ii) Let X  R m be definable and non empty, m > 1, and let 'If: Rm -+ R m - 1 1)(,
the projection on the first m -1 coordinates. Then 'If X  R m - I so we way
assume inductively that an element a == e('lfX) of 'If X has been defined.
Then Xa  R and we put e(X):== (a,e(Xa»).
(1.2) PROPOSITION (DEFINABLE CHOICE).
(i) II S  Rm+n is definable and 'If: Rm+n -+ R m the projection on the first 11/
coordinates, then there is a definable map I: 'If S..... Rn such that f(f)  S.
(ii) Each definable equivalence relation on a definable set X has a definable ,'('I
01 representatives.
PROOF. For (i), define I(x):== e(Sx) for xE'lfS. For (ii) note that
{e(A); A is an equivalence class}
is a definable set of representatives. 0
(1.3) COMMENT. Model-theorists will recognize in (i) the property of having defill
able Skolem functions. As to (ii), consider our structure g on R as a category \dh
the definable sets as objects and the definable maps between them as morphisliis.
If E  X x X is a definable equivalence relation on a definable set X  H"'.
then (ii) provides a definable map I : X -+ X such that E :::: kernel(f), that. is,
x Ey {:} I( x) :::: I( y), for all x, Y EX. Hence for each definable map g: X  f{"
such that xEy => g(x):::: g(y) for x,y E X, there is a unique definable map
g': I(X) -+Rn such that 9 == g' 0 f. Therefore the definable set I(X), together wit h
the map I:X -+/(X), serves as a quotient X/E in the category 8.
(1.4) We equip from now on Rm, m > 0, with the "supnorm" II:
!xl :::; maxi lXII, .. ., Ixml} for x == (XI, . .., x m ).
Note that then I I : R m -+ R is a definable continuous function. We also deli'l('
II:Ro:::: {O}R by 101 == o.
(1.5) COROLLARY (CURVE SELECTION). If a E cI(X) - X, where X is deJill llblr .
then there is a definable continuous injective map, ; (0, () -+ X, lor some ( > O.
such that limt-oO ,(t):::: a.
PROOF. Since a E cI(X) - X, the definable set {Ia - xl; x E X}  R contains
arbitrarily small positive elements, hence contains an interval (0, l), l > O. For each

POINT-SET TOPOLOGY IN O-MINIMAL STR1:CTURES
95
t in this interval there is x E X with la - xl = t. By definable choice there is then a
definable map,: (0, E).....X such that la - ,(t)i = t for all t E (0, E). By decreasing
( if necessary we may assume by the monotonicity theorem that, is continuous.
Obviously, is injective and lim;-.o ,(t) = a. 0
(1.6) Definable curve selection fails in the o-minimal structure (R, <) (see exercise
8 at the end of this section), so our standing assumption in this chapter that we are
dealing with an o-minimal expansion of an ordered group is appropriate here. A
point in the closure of a set X in a metric space is the limit of a sequence in X. This
fact is not available in our context, but curve selection offers a more than adequate
alternative (curves instead of sequences). We shall use curve selection in this way to
prove that the image of a closed bounded definable set under a continuous definable
map is closed and bounded. Here a set A  Rm is called bounded if lal < r for
all a in A and a fixed r E R. First some lemmas.
(1. 7) LEMMA. Let C be a bounded cell in Rm, m > 1, and 1!' : Rm ..... R m - 1 the
projection on the first m - 1 coordinates. Then 11' d(C) = cI(1!'C).
PROOF. We shall do the case C = (J,g)"c and leave the other case to the reader.
By the continuity of 11' we have 7rcl(C)  cI(1!'C). Let a E cI(1I'C). We have to
find s in R such tha.t (a, s) E cI( C). Of course we may assume a f/. 1!'C. Then
there is a continuous definable map, : (0, E) ..... 7rC such that lim;-.o ,( t) = a. We
are going to lift the curve, to a curve in C. Since C is bounded there is r > 0
such that -r < I(x) < g(x) < r for all x E 1!'C. Define the continuous function
>.: (O,E)..... R by A(t) := (J(-y(t») + g({(t»)/2. ote that -r < A(t) < r for all
t, so by the monotonicity theorem there is s E R such that lim;-.o A(t) = s. Then
t....... (t(t),>.(t»):(O,E).....C is a continuous definable map whose limit as t goes to 0
equals (a,s), so (a,s)EcI(C). 0
RBMARK. We cannot omit the hypothesis that C is bounded: in the ordered field
of real numbers, consider C:= {(x, 1/x)ER2: x> O}.
(1.8) Next we recall that if I:X.....y is a continuous map from a topological space
X into a Hausdorff space Y, then its graph f(J) is a closed subset of X X Y.
(1.9) LEMMA. Let I:X .....R n be a definable continuous map on a closed bounded
set X  Rm. Then I(X) is bounded in R n .
PROOF. Suppose 'Vt E R 3x EX I/(x)1 > t. By definable choice there is then a
definable map g:R.....X such that I/(g(t»)1 > t for all tin R. Since X is closed and
bounded it follows from the monotonicity theorem, applied to the m coordinates
of g, that limt-ooog(t) = x exists and belongs to X. So I(x) = I(limt-oog(t») =
lirnt-->oo/(g(t»), but the last limit cannot exist in R, since I/(g(t»1 > t for all t.
Contradiction. 0

96
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
(1.10) PROPOSITION. If f: X - RR is a continuous definable map on a closed
bounded set X  R m , then f(X) is closed and bounded in RR.
PROOF. By (1.9) we already know that f(X) is bounded in R n , so it suffice to
show that f(X) is closed. Consider Y := {(J(x),x) : x E X}  Rn+m (the
"reversed" graph of f) and write
Y = C 1 U ... U C k ,
where C!,..., Ck are cells in Rn+m. Since Y is closed by (1.8) we also have
Y = cI(Cl) U ... U cI(Ck)'
By (1.9) the set Y is bounded, so the cells C!, . . ., C k are bounded. Then it fo)]owh
from (1.7) that
1I'Y = 1I'cI(Ct)U"'U1I'cl(Ck)
= cI(1I'C 1 ) U ... U cI(1I'Ck),
where 1I':RR+m_RR is the projection on the first n coordinates. But 1I'Y = f(X),
so f(X) is closed in RR. 0
Here are three easy consequences, the first dealing with the case n = 1.
(1.11) COROLLARY. If f:X -R is a continuous definable function on a nonemply
closed bounded set X  Rm, then f assumes a maximum and a minimum value.
(1.12) COROLLARY. If f: X - RR is an injective continuous definable map on (!
closed bounded set X  Rm! then f is a homeomorphism from X onto f(X).
This last corollary perhaps requires explanation: f maps definable closed subs,,!s
of X onto closed subsets of f(X), by (1.10), hence it maps definable open subs"th
of X onto open subsets of f(X); now use the fact that an arbitrary open subset of
X is a union of definable open subsets of X.
(1.13) COROLLARY. Let f : X -+ RR be a definable continuous map on a closed
bounded set X  Rm and let Y = f(X). Then we have:
(i) A definable set S  Y is closed if and only if f-l (S) is closed;
(ii) A definable map g : Y -+ RP is continuous if and only if g 0 f : X - RP i,
continuous.
PROOF. Assertion (i) is clear from (1.10) and (ii) is an easy consequence of (j),
taking into account that for continuity of g it suffices that g-I(Z) is closed for all
definable closed Z  RP. 0
Closed bounded definable sets have a property that is like the completeness property
of projective varieties in algebraic geometry:

POINT.SET TOPOLOGY IN O-MI:-lIMAL STRUCTURES
97
(1.14) PROPOSITION. Let X  Rm be a closed botinded definable set and Y  Rn
(lny definable set. Then the projection map X x Y --> Y maps definable closed subsets
of X x Y onto definable clolled sublletll 01 Y.
pROOF. Let q:X x Y..... Y be the projection map, let A  X X Y be definable and
closed in X X Y, and suppose V E cly (q( A)). Then there is for each t > 0 in R a point
aEA such that Iq(a) - yl < t. By definable choice and the monotonicity theorem
tbis gives a definable continuous map a : (0, () ..... A, for some ( > 0, such that
Iq(a(t)) - yl < t for t E (0, (). Write a(t) = (,8(t), ,(t)), where ,8 and, = q 0 a are
dcfmable continuous maps from (0, () into X and Y respectively, so limt--() ,(t) = V.
Since X is bounded, it follows from Chapter 3, (1.6) that x := limt--() ,8(t) exists in
Rmj hence x EX, since X is closed. Then (x, V) E X x Y and limt--() a(t) = (x, V), so
(x, y) E A, because A is closed in X x Y. Therefore y = q( x, y) E q( A) as desired. 0
(1.15) EXERCISES.
in the first three exerdlles we consider a definable equivalence relation E  X x X
on a definable lIet X  R m .
1. Show that the definable set of representatives indicated in the proof of (1.2) is
definable in the model-theoretic structure (R, <, 1, +, E). (Recall that this set of
representatives is given by T:= {e(A): A is an equivalence class}.)
2. Show that E has only finitely many equivalence classes of dimension dim(X),
and that each of them is definable in the model-theoretic structure (R, <, 1, +, E).
3. Suppose all equivalence classes of E have the same Euler characteristic e. Show
that then the Euler characteristic of X is a multiple of e. (In particular, this
shows that for e > 1 there is no definable equivalence relation on R m all of whose
equivalence classes have exactly e elements.)
4. (Uniform continuity) Let X  R m be a closed and bounded definable set and
I: X ..... Rn a continuous definable map. Show that there is for each ( > 0 a 8 > 0
such that whenever Ix - vi < 8, x, y EX, we have I/(x) - l(y)1 < (.
5. (Fixed point theorem) Let X be a nonempty closed bounded definable subset of
Rm and I: X -+ X a definable map such that I/( x) - l(y)1 < Ix - yl for all distinct
points x, V EX. Show that 1 has a unique fixed point.
6. (Uniform curve selection) Let X  Rm be defina.ble. Show there are definable
maps (: ax -+ (0,00) and r : (0, () ..... X such that for each a E ax the function
t ..... r( a, t): (0, (a)) ..... X is continuous, injective and satisfies limt-+o r( a, t) = a.
7. Given a map I: A --> Rn, A S;; Rm, we callI locally bounded if each point
a E A has a neighborhood U in A such that I( U) is bounded. Let A  Rm be
definable and I: A -+ Rn definable. Prove the following equivalence:
f is continuous *=> f is locally bounded and r(f) is closed in A x R n .

98
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
Note that the local boundedness condition cannot be omitted in the reverse im-
plication <=: the function f:R..... R given by f(x) = l/x for x =I 0, f(O) = 0, is
definable in the ordered field of real numbers, has closed graph fU) in R 2 , but i
not continuous at O.
8. Consider the o-minimal model-theoretic structure (R, <) and the set
X := {(x, Y) E R2: 0 < x < 1, 0 < y < 2},
which is definable in (R, <) using the constants 0,1,2. Note that (1,2) E cl(X )
and show that there is no subset Y of X such that Y is definable in (R, <) using
constants, dim(Y) = 1 and (1,2) E cl(Y).
2. Fiberwise properties
(2.1) Strictly speaking the results of this section are not needed later in this book.
However, for those familiar with elementary model theory this section provides a
very cheap way to obtain the triviality theorem of Chapter 9 from the triangulation
theorem of Chapter 8, see Chapter 8, (2.13) and (2.14), for more details. The proof
of the theorem below nicely illustrates the use of the definable choice principle.
(2.2) THEOREM. Let S  Rm+R be a definable set such that for each x E Rm thr
fiber S" is on in RR. Then there is a partition of Rm into cells Gl"." Ck such
that S n (G. X RR) is on in C. X Rn for i = 1,. .., k. (Same for "closed" instmd
of "on".)
PROOF. By induction on m. The case m = 0 is trivial. Let m> 0 and assume the
result holds for lower values of m. First we show:
(*)
{ Each open cell G in R m has an open subcell D
such that S n (D x R n ) is open in D x R n .
To see this, note that G = G(l) U C(2) with definable C(i) as follows:
G(l).- {xEG: ({x}xS,,)ncl(GxR n )-S)=l0},
C(2).- {XEG: ({x}xS,,)ncl(GxR R )-S)=0}.
If C(2) has nonempty interior, we can take for D any open cell contained in C(2).
Now assume C(2) has empty interior. Then G(I) has nonempty interior, and rcvlac.
ing G by an open subcell contained in C(I) we reduce to the case that G = C(1).
So for each x E G there is a point s(x) E S" with (xAx») E cl ( C X RR) - S). Sincp
S" is open in RR there is also for each x EGan £(x) > 0 such that if y ERn and
Iy - s(x)1 < (x), then yES". By definable choice we may take x ...... s(x): G..... R"
and x f--+ £( x) : C ..... R to be definable functions: replacing G by a suitable open
subcell we may further reduce to the case that these functions are continuoub.

POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
99
Then {(x,y) E ex Rn: x E C, Iy - s(x)1 < f(X)} is a subset of S, and open in
ex R"; it also contains the points (x,s(x») (xEC), which belong to the closure of
(C x R n ) - S, a blatant contradiction. This proves (*).
I'i ow let A be the union of all open boxes D in Rm such that S n (D x R n ) is open
in D X Rn. Then A is definable, S n (A x R n ) is open in A x Rn, and (*) implies
that dime Rm - A) < m. Write R m - A as a disjoint union of cells Bl,. .., Br where
each B j has dimension < m.
Let B be one of the cells B i , and let dim(B) = d < m. Then p(B) is an open
cell in R", and we can use the homeomorphism (x,y)...... (pB(x),y):B X R"-+
p( B) x R n and apply the inductive hypothesis to the i!llage of S n (B x R n ) under
this homeomorphism. The "closed" version follows by pM sing to complements. 0
(2.3) COROLLARY. Let S'  S be definable sets in R m +", and let A  R m be
definable such that S is open in Sa for all a E A. Then there is a partition of A
into definable subsets Al, . . . , AM such that S' n (Ai X Rn) is open in S n (Ai x R"),
for i = 1, . . ., M. (Same with "closed" instead of "open".)
PROOF. In this CMe it is more convenient to do the closed version. Replacing S by
S n (A x Rn) we may as well assume that the projection map Rm+n - Rm maps S
into Aj thus S is closed in S., for each x E R m . Let S. be the subset of Rm+n with
8; = cJ( S) for each x E Rm. Then S. is definable and S; n s., = S for x E Rm.
Now apply the closed version of the theorem above to S.. 0
(2.4) COROLLARY. Let S  R m + n be definable, f : S ..... R k a locally bounded
definable map, and A  Rm a definable set such that for all a E A the map fa :
Sa_R k is continuous, where fa(Y):= f(a,y). Then there is a partition of A into
definable subsets Al' . . ., AM such that each restriction
flsn (Ai X Rn):Sn (Ai X R")-+R k
is continuous.
PROOF. Apply the closed version of (2.3) to the definable subset f(J) of S x Rk,
noting that r(f)a = r(fa) is closed in Sa X R k = (S X Rk)a for each a E A.
This gives a partition of A into definable subsets Al"'" AM such that for each
i E {I,.. .,M} the set r(J) n (Ai X Rn+k) = r(JIS n (Ai X R n ») is closed in
(S x Rk) n (Ai x R"H) = (S n (Ai X R"» x R k . Hence, by (1.15), exercise 7, the
restrictions flS n (Ai X Rn) are continuous. 0
(2.5) EXERCISES.
1. Assume (R, <,8) expands an ordered field. Show that then (2.4) holds even
without the assumption that f is locally bounded.

100
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
2. Assume (R, <, S) expands an ordered field. Let S  Rm+" be definable, f :
S - R k a definable map, and A  Rm a definable set such that flS n (A X R") i>
injective and la: Sa - R k is a homeomorphism from Sa onto fa(5.) for all a E.1.
Show that there is a partition of A into definable subsets AI,"" AM such th<l t
each restriction liS n (Ai X R"): S n (Ai X R") - R k is a homeomorphism frotll
S n (Ai X R") onto I(S n (Ai X R"»).
3. Paths and partitions of unity
The two topics of this section are unrelated. The results here will be essential ill
several later chapters.
(3.1) DEFINABLE PATHS.
Let X  R m . A definable path in X is a definable continuous map I: [a,b]- R'"
with a, bE R, a < b, taking values in X. Such a path I is said to connect th('
points I(a) and I(b). IfI:[a,bJ-X and 8:[b,c]_X are definable paths iu X
with I(b) = 8(b) we have a definable path I V 8: [a, c] -+ X in X agreeing on [a, b:
with I and on [b, c] with 8. Using this concatenation construction it is clear that
if the points x,y E X can be connected by a definable path in X, and also tht'
points y, z EX, then x, z as well. It is easy to see that if X is definable and evE'rY
two points in X can be connected by a definable path in X, then X is definably
connected. The converse fails for the o-minimal model-theoretic structure (R. <)
(see the exercise at the end of this section). It does hold however under our standinr;
assumption in this chapter that we are dealing with an o-minimal expa.nsion of an
ordered abelian group:
(3.2) PROPOSITION. Suppose the definable set X is definably connected. Then any
two points in X can be connected by a definable path in X.
PROOF. Assume first that X is a cell, without loss of generality an open cell in li"'.
This case is handled by induction on m. The case m = 1 is obvious. For m > 1,1('1
C be the projection of X in Rm-I, so X = (J, g) with f, 9 functions on C, and I<'t
(y,r) and (z,s) be two points in X, y,zEC. Assume I and 9 are R-valued. ('1'1.('
case that 1= -00 or 9 = +00 is left to the reader.)
We first connect (y,r) by a "vertical" path in X to (y,(J(y) + g(y»/2); by th,'
inductive hypothesis there is a definable path I: [a, b] - C connecting y to z and
this path lifts to the path la, b] -+ X given by t -+ (r(t), (I( l(t») + g("«t»)/'2).
connecting (y, (f(y) + g(y»)/2) to (z, (f(z) + g(z»)/2); the last point can in turn
be connected to (z,s) by a vertical path in X. Concatenating the three pathH wc
obtain a definable path in X connecting (y,r) to (z,s).
In the general case, we use Chapter 3, (2.19), exercise 5 to write X as a union of
cells CI,. . . , C k where for each i < k, either C; intersects t he closure of CHI, or

POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
101
(;+1 intersects the closure of Ci. Then by curve selection there is a definable path
in Ci U CHi that connects a point of Ci with a point in CHi. Now combine this
with the fact that the desired result has already been proved for cells. 0
One particular consequence of (3.2) will be used in Chapters 8 and 9 in proofs of
triangulation and trivialization:
(3.3) COROLLARY. Let X and Y be definable sets in R n with X definably con-
nected, and X n bd(Y) = 0. Then either X  Y or X n Y = 0.
PROOF. If,not, there would be points x E X - Y and y E X n Y. Take a definable
path,: [a,b] -+ X connecting x to y. Put c := inf{t E [a,b]: ,(t) E V}. Then
j'(c)EX n bd(Y), contradiction. 0
PARTITIONS OF (;NITY.
(3.4) LEMMA. Let A  B  Rm be definable sets, with A closed in B. Then there
is a definable continuous junction f: B -+ [0,1] with f-l (0) = A.
PROOF. Assuming A =10, take f(x) = min(l,d(x, A», where
d(x,A)==inf{lx-al: aEA}. 0
(3.5) LEMMA. Let Ao and Ai be disjoint definable closed subse18 of the definable
set B  R m . Then there are disjoint definable open subsets Uo and Ui of B with
Ai <;:: Ui, i = 0,1.
PROOF. Take functions fo and It as in (3.4), and put
U o := {XEB: fo(x) < hex)}, and Ui :== {XEB: h(x) < fo(x)}. 0
(3.6) LEMMA (SHR.INKING OF OPEN COVERINGS). Let the definable set B  R m
be the union of its definable open subsets U 1 , . . . , Un. Then B is also the union of
definable open subsets Vi"'" V n with clB(V;) <;:: Ui for i == 1,..., n.
PROOF. Assume inductively that Vi  Ui has been defined for i == 1,. .., k (k < n)
such that V; is definable, open in B, clB(Vi)  Ui and Vi,'''' Vk, Ui:+i," ., Un
cover B. Then apply (3.5) to the following two disjoint closed subsets of B:
Ao :== B- Uk+1,
Ai := B- ( U V; u U U; ) . 0
i=l j=k+Z

102
POINT-SET TOPOLOGY I:-l O-MINIMAL STRUCTURES
(3.7) LEMMA (DEFINABLE PAR.TITION OF UNITY). With Ul,...,U n and B as in
(9.6), there are definable continuous functions It,. ..,fn:B....[O' 1] such that
(i) supp(fi)  Ui for i = 1,.. ., n, where SUPp(fi) := clB ({ x E B: Ji(x) =I O}),
(ii) :Efi(x) > 0 for all xEB.
Moreover, if (R, <, S) expands a real closed ordered field, thm (ii) can be replacol
by :E fi = 1. (We then call It, . . ., fn a definable partition of unity for the covering
U 1 ,...,U n .)
The easy proof using (3.6) and (3.4) is left to the reader.
(3.8) LEMMA. Suppose (R, <, S) expands an ordered real closed field. Let Ao awl
Al be disjoint definable closed subsets 0/ a definable set B  R m . Then there is n
continuous definable/unction /:B....[O,I] with /-1(0) = Ao and /-1(1) = Al.
PROOF. Choose Uo and lI 1 as in (3.5), and apply (3.7) to get a definable partition
of unity fo,It,1z for the covering Uo,Ul,B - (Ao U Ad of B. Then use (3.4)
to get definable continuous functions 90 : B .... [0,1/2] and 91 : B ..... [1/2,1] wit h
9i 1 (0) = Ao, 91 1 (1) = A 1 . Now set f:= f090 + f191 + (1/2)/2. 0
(3.9) EXERCISE. Show that in the model-theoretic structure (R, <, 0, 1,2) the d<'
finable set {(x,y): 0 < x < 1, 0 < y < 2} U {(1,2)} in R2 is definably connected
but not definably path connected (the latter defined as in (3.1».
4. Curves, proper maps, and identifying maps
This section contains miscellaneous results used only in the last chapter of this
book. The reason for including these results here is simply that they hold under
weaker assumptions than are in force later on in this book.
(4.1) DEFINITIONS. Let X  R m be a definable set.
(1) A definable curve in X is a definable map,: I ..... X, for some inter viti
I = (a, b) in R. We do not require, to be continuous, and we allow of
course that b = +00. Our interest is in the behavior of I near its right
endpoint b, that is in the germ of I at b; we note that I will be continuous
on some smaller interval (a', b) with the same right endpoint. (That we an'
only interested in the behavior near the right endpoint is just a convention.
We could as well have chosen the left endpoint.)
(2) Let ,:(a,b)....X be a definable curve in X. Given a point PERm we writ('
, -+ p as shorthand notation for limt-ob I( t) = p. (We do not require that
PEX.)
Let X  Rm be definable and, a definable curve in X. We call, completable jf
there is a (necessarily unique) point PERm such that I--+P. If ,.....PEX, then we
call, completable in X. Here are some simple observations.

POINT-SET TOPOLOGY II\' O-MINIMAL STRUCTURES
103
(i) X bounded =? {is completable. (By Chapter 3, (1.6).)
(ii) X is closed and bounded =? (is completable in X.
(Hi) If f: X -+ Y is a definable map into a definab1e set Y  Rn, then f(r) := f o{
is a definable curve in Yi if f is in addition surjective (J(X) = Y), then
each definable curve /3 in Y can be lifted to a definable curve a in X, that
is, f(a) = /3. (By definable selection.)
(iv) {is either injective on a subinterval (a', b), or constant on such a subinterval.
(By the monotonicity theorem.)
(4.2) LEMMA. Let f: X -+ Y be a definable map from the definable set X  R m
into the definable set Y  Rn, and let p EX. Then f is continuous at the point p
iff for each definable curve { in X with (-: p we have f( () -+ f(p).
PROOF. If f is continuous at p, the "curve continuity" follows immediately. Sup-
pose f is not continuous at pj so there is € > 0 such that the definable set
{Ix - p:: x E X, lJ(x) - f(p)1  £} contains arbitrarily small positive elements, so
it contains an interval I with left endpoint O. By definable selection this produces
a definable curve (: I -+ X such that i{(t) - pi = t and If(l(t») - f(p) I  € for
t E J. Then its "transform" {': -J -+ X given by {'( -t) = {(t) has the property
that (' -+ p but not f( I') -+ f(p). 0
REMARK. As the proof shows, it suffices to consider just injective definable curves
{ in the statement of the lemma.
(4.3) LEMMA. Let the definable curve /3: (a, b) -+ Rn be completable, /3 -+ p, and
let £ > 0 be such that a < b - £ < band /3 is continuous on [b - £,b). Suppose
Q: J -+ /3([b - £, b» is a definable curve in /3([b - €, b») not completable in /3 ([b- £, b»).
Then a is a reparametrization of /3 near b: there are a subinterval I' of I with
the same right endpoint a8 I, an a' E (b - £, b) and a definable strictly increasing
homeomorphism h :1' -+ (a', b) such that o:(t) = /3 (h(t») for tE I'.
PROOF. The curve a is necessarily completable in the closed and bounded set
,B ([b - £, b») U {p}. Hence 0: -+ p, and a and /3 are both injective near b. Thus by
decreasing the interval I, keeping the same right endpoint, and also decreasing f,
we may as well assume that a is continuous and injective, and /3 is injective on
[b - £, b), and pit /3 ([b - £, b». Write 0: = (a1,' .., an) and /3 = (/31,.. . , /3n). By
further shrinking we may also assume one of the /3i is strictly monotone on [b-£,b),
say /31 is strictly increasing on [b - £, b). Since 0:1 (I)  /31 ([b - £, b», the function
Q1 is also strictly increasing near the right endpoint of I.
Now use exercise 2 from Chapter 3, (1.9) to conclude that there are a subinterval
!' of I with the same right endpoint as I, and a' E (b - £, b) and a definable strictly
mcreasing homeomorphism h: I' -+ (a', b) such that 0:1(t) = /31 (h(t») for t E I'.
Then it follows easily from a(I)  /3([b - £, b») and the injectivity of /31 on [b - £, b)
that also a(t) = /3(h(t») for tEl'. 0

104
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
(4.4) DEFINITION. Let I:X _Y be a definable continuous map from the definable
set X  Rm into the definable set Y  RR.
(1) We call I definably proper if for each definable set K  Y we have; K is
closed and bounded in RR ::::> rl(K)  X is closed and bounded in Rm,
(2) We call I definably identifying if I is surjective U(X) = Y), and for
each definable set K  Y we have; 1-1 (K) is closed in X ::::> K is closed
in Y.
If X  Rm is a closed and bounded definable set and I ; X _ RR is a definabl('
continuous map, then I is obviously definably proper, and, as a map from X onto
I(X), also definably identifying, by (1.10). A more typical example of a definably
proper map is a projection map R m x Y - Rm, where Y  RR is a closed and
bounded definable set. These notions, and the lemmas that follow, will becon1(,
especially useful in the last chapter when we construct quotient spaces of definable
sets modulo definable equivalence relations.
(4.5) LEMMA. Let I;X _y be as above: X,Y definable, I definable and contin-
uous. Then
(1) I is definably proper iff each definable curve I in X whose image l('"t) i8
completable in Y is completable in X,.
(2) I is definably identilying iff each definable curve f3 in Y that is completabh>
in Y lifts to a definable curve a in X that is completable in X.
PROOF. (1) Suppose I is definably proper, I : (a,b) -+ X is a definable curve in
X and 1(1) is completable in Y, say I( 'Y) -+ y E Y. Take a' E (a, b) such that '"'{ is
continuous on [a', b). Then the definable set
K;=U('"t(t»: tE[a',b)}U{Y}Y
is closed and bounded in RR, so 1- 1 (K)  X is closed and bounded in Rm, 50 I
must be completable in 1- 1 (K), hence in X.
Conversely, suppose I is not definably proper; then there is a definable f(  )'
that is closed and bounded in RR, such that I-I (K) is not closed in Rm or not
bounded. If 1- 1 (K) is not bounded, we can by definable selection choose a definable
curve I in 1- 1 (K) that is not completable in Rm. Then lCi) is a curve in fI',
hence completable in K, and so in Y. If 1- 1 (K) is not closed in R m , take ]J E
cI(!-l(K») - 1- 1 (K), and use curve selection to get a definable curve 'Y in 1- 1 (/1')
such that 'Y -+ p. Since 1- 1 (K) is closed in X, we have prf. X, so 'Y is not completabk
in X. But, as before, f( I) is completable in Y.
(2) Suppose I is definably identifying, and let f3; ( a, b) -+ Y be a definable cur\'(' ill
Y that is completable in Y. We have to lift /3 to a curve a in X that is cornpletabk
in X. We may assume f3 is injective and continuous on a subset [b- £,b) of (a, b),
( > 0, since otherwise /3 would be constant on such a set, and we can then use the
fact that I is surjective.

POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
105
After further decreasing £ we may also assume that /3 -+ y E Y - ,8([b - £, b»), so
a([b - £,b)) is not closed in Y. Hence f-l(,8([b - £,b»)) is not closed in X. By
curve selection there is a definable curve a in f-l (/3 ([b - £, b»)) with a -+ X and
x E X - f-l(,8([b - £,b))). Then f(a) is a definable curve in {3([b - £,b)) with
j(a) -+ f(x) and f(x)  ,8([b - £,b»). Hence by lemma (4.3) above, f(a) is a
reparametrization of,8 near b. Reparametrizing a we achieve that f( a) = ,8.
Conversely, suppose f is surjective but not definably identifying. Let K !;; Y be
definable such that rl(K) is closed in X but K is not closed in Y. Then there is a.
definable curve ,8 in K with ,8 -+ Y E Y - K. If,8 were to lift to a definable curve a in
X that is completable in X, say a-+xEX, then image(a)!;; r 1 (K) and f-l(K)
would be closed in X, so xEf-l(K), hence y = f(X)EK, contradiction. ,0
(4.6) COROLLARY. If f: X -+ Y as above is definably proper and surjective, then
f is definably identifying.
This is immediate from (4.5).
(4.7) COROLLARY. Let f : X -+ Y as above be surjective. Then f is definably
identifying iff for each definable K !;; Y we have
f(clx(!-I(K»)) = cly(K).
PROOF. Suppose f is definably identifying, and let K !;; Y be definable. By
continuity of f we have f(cl x (!-I(K»)) !;; cly(K). For the reverse inclusion,
let y be in the closure of Kin Y, and take a definable curve ,8 in K with /3 -+ y.
Lift ,8 to a definable curve a in X with a -+ X EX, so that f( x) = y. Then a lies
in f-l(K), so XEcl x U- 1 (K»), hence y = f(x)Ef(cl x (!-I(K»)).
Conversely, assume each definable K  Y has the property mentioned in the corol-
lary. Let ,8: (a, b) -+ Y be a definable curve such that ,8 -+ y E Y. We have to lift ,8
to a defina.ble curve in X completable in X. We may assume that /3 is not constant
near b, hence we may assume /3 is injective and continuous on a subset [b - £, b)
of (a, b), £ > 0, and y  K := ,8([b - £, b»). Then y E cly(K), so there is an x in
cl X (!-I(K») with f(x) = y. Let a be a definable curve in r 1 (K) with a-+x.
Then f(a) is a definable curve in K, hence the curve f(a) is a reparametrization of
i3 near b, by (4.3). Then a can be similarly reparametrized so that f(a) =,8. 0
(4.8) EXERCISES.
1. Let f: X -+ Y be a definably proper map. (This includes the assumption that X
nd Yare definable sets and f is definable and continuous.) Show that if A!;; X
IS definable and closed in X, then f(A) is closed in Y. Show that if !':X'-+Y' is
a definably proper map, then f x f': X x X' -+ Y X Y' is definably proper. Show
that if g: Y ..... Z is definably proper, then 9 0 f : X -+ Z is definably proper.

106
POINT-SET TOPOLOGY IN O-MINIMAL STRUCTURES
2. Let (R, <, S') be an o-minimal structure with S  S' and let f; X ..... Y be <1.
definable continuous map between definable sets X and Y in R m and RR, Wh0n'
"definable" is taken in the sense of S. Assume also that Y is locally closed in R'.
Show
f is definably proper with respect to (R, <, S)
$}
f is definably proper with respect to (R, <, S').
In the next exercise we assume that (R, <,8) = (R, <,8) is an o-minimal expansio:1
of the ordered additive group of reals. We recall that a continuous map f; X..... }'
between Hausdorff spaces X and Y is called proper if f maps closed subsets 0:
X onto closed subsets of Y and each point y E Y has compact fiber f-l(y). 'fhi,
definition agrees with that of Bourbaki [6, Ch. 1, 10, Thm. 1] who also shows that
if X and Yare Hausdorff spaces and the continuous map f:X..... Y is proper, th('n
f-l(C) is compact for each compact C  Y.
3. Let f; X ..... Y be a definable continuous map between definable sets X £;; R"
and Y  RR. Show that f is proper if and only if f is definably proper.
Notes and comments
The definable choice principle of Section 1 combines the model-theoretic propcrti('s
of "definability of Skolem functions" and "elimination of imaginaries". The first "('
of definability of Skolem functions in the semialgebraic case seems to be in Delz(':1
[14]. The stronger principle of definable choice was pointed out for real closed fipld,
in [18]. That definabillty of Skolem function implies curve selection (for real and
p-adic fields) was observed by Scowcroft and Van den Dries in [52j. Brumfiel m,lk('s
the heuristically useful remark in [8] that curves play in semi algebraic geom('( 1',1'
the same role as sequences in metric spaces. Peterzil and Steinhorn show in [H]
that proposition (1.10) on images of closed bounded definable sets remains tru0 for
arbitrary o-minimal structures.
Bochnak, Coste, and Roy prove the fiberwise properties of Section 2 in the 5('::1:
algebraic case in their book [4] by methods that seem specific to that situlltioi:.
After the more general theorem (2.2) a.bove had been found, Speissegger [57] estab-
lished these properties for arbitrary o-minimal structures. The "partition of lI:ii:,I"
results" of Section 3 are taken over from the semialgebraic case treated by Iklf>
and Knebusch in [13, 1]. Similarly I adapted from Brumfiel [8] the material in
Section 4 on curves, proper and identifying maps.

CHAPTER 7
SMOOTHNESS
Introd uC,tion
In this chapter we fix an o-minimal structure (R,<,S) that expands an ordered
field (R, <,0, 1,+, -,.). Recall from Chapter 1, (4.6) that this ordered field is then
necessarily real closed. In Section 1 we discuss differentiability, in Section 2 we
show that definable functions in one variable are piecewise differentiable and that
definable Cl-maps in several variables satisfy the implicit function theorem. This
allows us to obtain in Section 3 the decomposition of definable sets into "smooth"
cells. The smooth cell decomposition is then used in the last section to prove the
good directions lemma, an important tool in triangulating definable sets in the next
chapter.
1. Differentiability in ordered fields
(1.1) We start by recording some elementary calculus facts valid over arbitrary
ordered fields (R,<,O,l,+,-,.). (o o-minimality assumptions or definability
assumptions on sets and functions are needed in this section.) To shorten no-
tations we denote this ordered field just by R. We equip R with the interval
topology, and each set Rm with the corresponding product topology, and we put
Ixl := sup{lxil: i = 1,..", m} for x = (Xl,..., x m ) E R m . We also equip R m with
the usual dot product: x. Y = XIYl +... + XmYm' In this section we omit proofs
since they are the same as for the field of real numbers.
(1.2) DEFINITIO:-;. Let I  R be open. A function f : I -+ R n is said to be
differentiable at a point x E I with derivative a E R'" if
limr1(f(x+t)-f(x») = a.
!-oO
Note that then f is continuous at x and that a is unique; we write a = f'(x).
(1.3) Let the functions f, 9 : 1-+ R n be differentiable at x. Then the sum f + 9 and
the dot product f. 9 are differentia.ble at x a.nd
(J + g)'(x) = f'(x) + g'(x),
(J.g)'(x) = f'(x).g(x)+f(x).g'(x).
107

108
SMOOTHNESS
If moreover n = 1 and 9 does not vanish on I then 1/ 9 is differentiable at x and
U/g)'(x) = (f'(x). g(x) - I(x)' g'(x»)/i(x).
COll6tant maps are differentia.ble everywhere with derivative O.
The identity function on R is differentiable everywhere with derivative 1.
Let I, J be open subsets of R, let I: 1-+ R be continuous and differentia.ble at till'
point x E I. g: J -+ R differentiable at I( x) E J. Then 9 0 I, defined on the open spt
In 1- 1 (J), is differentiable at x and
(g 0 I)'(x) = g'(f(x»). I'(x).
(1.4) DIRECTIOI>AL DERIVATIVES. Consider now a map I:U _R n with U  IC"
open, a point x E U, and a vector v E R m . We say that I is differentiable at .
in the v-direction with derivative a E R n if the function t 1-+ I(x + tv) (d"fillpd
on an open neighborhood of 0 E Rrn) is differentiable at 0 with derivative a, that is
lim C 1 (l(x + tv) - I(x») = a.
-
In that case we write dxl(v) = a. Let e(l),. ..,e(rn) be the standard basis of the
R-linear space Rm, so e(l) = (1,0,...,0), etc. As in the case of the real field we
write (8f/8xi)(X) for dxl(e(i» and call it the i 1b partial of I at x.
(1.5) THE DIFFEREI>TIAL OF A MAP. Let I = (11,..., In): U -+ Rn be a map Oil
an open subset U of Rm. Let x E U and T: Rm -+ R n be a linear map.
We call I differentiable at x with differential T if for each ( > 0 we havp
If(x + v) - I(x) - T(v)1 < (lVi,
for all sufficiently small vectors v E Rm. Then I is continuous at x and '/' is
unique; we write dxl = T. This notation is consistent with that in (104). sillc!'
if T is as above, then I is differentiable at x in the v.direction for each 11, and
T( v) = directional derivative of I at x in the v-direction. For rn = 1 this notioll of
differentiability coincides with the one in (1.2), with dxl(l) = I'(X).
The map f = (f1,... ,in) is differentiable at x iff each coordinate function Ii: U -.11
is differentiable at x. In that case all partials (81;/ 8x j)( x) exist and the n x 1/1
matrix (81;/8xj)(x) is exactly the matrix of dxl relative to the standard basI's. It
is called the Jacobian matrix of f at x.
(1.6) BEHAVIOR oNDER ALGEBRAIC OPERATIONS. If the maps I,g: U -, R" Mt'
differentiable at x E C;, U open in Rrn, then I + 9 and cf, for each c E it, a \'e
differentiable at x and
dxU + g)
dxcl
dxf + dxg,
= c. dxf.

SMOOTHNESS
109
[(also h;V -RP is differentiable at f(X)EV, V open in RR and f continuous, then
h 0 f, defined on un f-1 V, is differentiable at x and
d",(h 0 f) = (df(x)h) 0 d",f.
Each R-linear map R m -+ RR is differentiable at each point in Rm with itself as
differential.
2. Inverse function theorem
(2.1) COr.;VENTION FOR THE REST OF THIS CHAPTER.
'R = (R,<,8) is an o-minimal expansion of an ordered field (R,<,O,l,+,-,.).
(This ordered field is then in fact real closed by Chapter 1, (4.6).) We let Ilxll :=
(x. x )1/2 denote the usual 'euclidean norm' of the vector x E Rm.
(2.2) LEMMA (ROLLE). Suppose a < b in R and the junction f: [a,b] -+ R is
definable, continuous, f(a) = f(b), and f is differentiable at each point of (a,b).
Then f'(C) = 0 for some eE(a,b).
PROOF. Take c, a < e < b, such that f(c) is maximal or minimal. Then clearly
f'(e) = O. 0
This has the following easy consequences.
(2.3) MEAN VALJ:E THEOREM. Suppose a < b in R, f:[a,bJ-R is definable and
continuous, and differentiable at each point of (a, b). Then for some c E (a, b), we
have f(b) - f(a) = (b - a) . f'(c).
(2.4) THEOREM ON CONSTANTS. Under the same assumptions as in the mean
value theorem, suppose that moreover f'(x) = 0 for all x E (a, b). Then f is constant.
In the following I denotes an interval. Our first goal is to prove
(2.5) PROPOSITION. If f : I -+ R is definable, then f is differentiable at all but
finitely many points of I.
This requires several Jemmas.
LEMMA 1. Let f: I -+ R be definable. Then for each x E I the limil8
f'(x+) .- limc 1 (J(x+t)-f(x)},
t!O
f'(x-) .- !im c 1 (J(x + t) - f(x)}
ITO

110
SMOOTHNESS
exist in Roo. II moreover I is continuous and I'(x+) > 0 lor all x, then I is strictly
increasing and its inverse 1-1; 1(1) -+ R satisfies (1-1)' (y+) = 1/ 1'( x+), lor x EO I
and I(x) = yE/(1). (Here 1/ + 00 := 0.)
PROOF. For fixed x the function g: t...... (I(x +t) - I(x))/t, defined on an interval
(O,€),is definable, whence the limit I'(x+) = limHOg(t) exists by Chapter 3, (l.fi).
Similarly for 1'( x-).
Suppose now that I'(x+) > 0 for all x, and I is continuous. If I were not strictly
increasing then I would be constant or strictly decreasing on some subinterval.
contradicting I'(x+) > 0 on that subinterval. The formula for the right derivative
of 1-1 is obtained in the usual way.
LEMMA 2. Let I; I -+ R be definable and continuous, and suppose the maps J: .-
I'(x+) and x...... I'(x-) are R-valued and continuous on I. Then I is differentiflblr
at each point 01 I, and I': I -+ R is continuous.
PROOF. It suffices to show that 1'( a+) = 1'( a-) for all a E I. Suppose the contrary,
say I'(a+) > I'(a-) for a certain a E I. Then there are c E R and a subinterval J
of I around a such that I'(x+) > c> I'(x-) on J. Hence the definable functin!
g; J -+ R given by g(x) ;= I(x) - cx has the property that g'(x+) > 0, g'(x-) <
o for all x, so 9 would be both strictly increasing and strictly decreasing on J.
Contradiction.
LEMMA 3. Let I: I -+ R be definable. Then there are only finitely many x E I 6'Il"/t
that l'(x)E{-oo,+oo}.
PROOF. Suppose the definable set {XEI: I'(x+):;::: +oo} is infinite. Then thb
set contains a whole interval, and for the sake of deriving a contradiction we IIIay
as well assume that I'(x+) = +00 for all x E I and that I is continuous. By lernIII<t
1 this implies that I is strictly increasing, hence 1'( x-)  0 for all x E I.
After further shrinking the interval we may also assume that we are in one of the
following cases;
(i) I'(x-) = +00 for all x in I,
(ii) I'(x-)E R for all x in I, and the map x...... I'(x-) is continuous on I.
In case (i) the inverse of I satisfies (1-1 )'(y-) = (1-1 )'(y+) = 0 for all y E f(T),
so that by the theorem on constants 1-1 is constant, contradicting the injectivity
of 1-1. In case (ii) we can apply the same argument as in the proof of lemma 2 to
get a contradiction.
PROOF OF (2.5). By the monotonicity theorem and lemma 3 we can reduce to
the case that I is continuous, and I'(x+) and I'(x-) are R-valued and continuouS
functions of x E I. Now apply lemma 2. 0

SMOOTHNESS
111
(2.6) CONTINUOL'S DIFFERENTIABILITY.
In the following definition and lemmas we denote by I = (/1,"" In): V -+ Rn a
definable map on a (definable) open set V  R m . We also put
Lin(Rm,R n ) := R-linear space of R-linear maps R m -+ R n .
DEFINITION. We call I a Cl-map if the partials (81;/fJxj) are defined as R-valued
functions on V and are continuous. (It follows that I is continuous.)
LEMMA. II I is C l , then I is differentiable at each point of V, and the map
x....... d,J: V -+ Lin(R m , R n )  R nxm
is continuous. Conversely, il I is differentiable at each point 01 V and the map
x ....... dx/: V -+ R nxm is continuous, then f is a Cl-map.
(Here we identify each R-linear map from R m to R n with its n x m matrix relative
to the standard bases.)
The second part of the lemma is an immediate consequence of the fact stated at the
cnd of (1.5). The proof of the first part follows the classical proof, which depends
on the mean value theorem. (That is why we restrict ourselves to definable maps,
for which this result is available.)
Next we define the norm of a linear map.
(2.7) DEFINITION. For an R-linear map T:Rm-+R n we put
ITI := max{ITxl: Ixl $ 1, XER m }.
(f'ote that the maximum exists since T is definable and continuous and that
IT(x)1 $ITI.lxl for all xER m .)
(2.8) LEMMA. Let I: V -+ R n be Cl, and let [a, b] := {(I - t)a + tb: 0 $ t $ I}
be a line segment contained in V. Then
lJ(b) - l(a)1 $ Ib - al' max ;dyfl.
y E [a,b]
PROOF. Define g: [0, 1] -+ R n by get) := 1((1- t)a + tb). Then g is differentiable at
each t, 0 < t < 1, with g'(t) = dyl(b - a), where y := (1 - t)a + tb, so Ig/(t)1 $ M,
with
M := Ib- al. max dyl.
y E [a,b]
Then dearly by the mean value theorem, If(b) - l(a)1 = Ig(l) - g(O)1 $ M. 0

112
SMOOTHNESS
(2.9) LEMMA. With the same assumptions as in the previous lemma, let x E U.
Then
If(b) - f(a) - drf(b - a)1  [b - al' max idf - dxfl.
 E [.,bl
PROOF. Apply the previous lemma to the map y...... fey) - dxf(y). 0
(2.10) LEMMA. With the same assumptions, let m = n, a E U and suppose d.l i"
invertible. Then there are f > 0 and c > 0 in R such that
If(x) - f(y)1  c/x - yl for all x, yE U with Ix - ai, Iv - al < f.
In particular f is injective on a neighborhood of a.
PROOf, Take f > 0 so 6mall that the open ball B(a,) is contained in U. I.e(
x,YEB(a,f). Applying the previous lemma, we have
If(x) - fey) - d.f(x - y)1  Ix - yl. max Id.f - d.fl,
. E [r,yJ
so
11(1') - f(Y)1  Id.f(x - y)l- Ix - yl' max Id.f - d.fl.
. E Ix,y]
Since d.f is invertible there is c' > 0, independent of x, y, such that
Id.f(x - y)[  c'lx - yl.
(Use Izi  c'/T(z)! for T ;;:: (d.J)-l, Z = daf(x - y).) Decreasing f if necessary we
may also assume that
Idb! - d.fl < c'/2 for all bEB(a,E).
Hence
If(x) - fey)!  c'lx - yl- (c'/2)lx - yl
 clx - yl
for c = c'/2. 0
(2.11) INVERSE Ft'NCTIO:-l THEOREM. Let f: U -+ Rm be a definable C1_mnp on
a definable open set U  R m and a E U a point where d.!: R m -+ R m is invErtible.
Then there are definable open neighborhoods U '  U of a and V' of f( a) such thnt
f maps U' homeomorphicallv onto V' and f-1 : V' -+ U' is also C1.
PROOF. In the statement of lemma (2.10) we may of course replace the norm 1.1
by the equivalent euclidean norm 11'11, which is more convenient here. Take (, C > 0
6uch that
111' - all  f =} x E U and dxf is invertible,
and 1/1' - all, lIy - all  f =} IIf(x) - f(y)1I  cllI - yll.

SMOOTHNESS
113
In particular, IIx - all = e =? IIf(x) - f(a)1I  c£. We claim that
{y: Ily - f(a)1I < (1/2)ce}  {f(x): Ilx - all < e}.
To see this, let lIy - f(a)1I < (1/2)c£ and consider the function
P(x) := Ilf(x) - yll2 = ]/;(x) - y;)2 on the balllix - all :5 {.
By Chapter 6, (1.11) the function P assumes a minimum value, but if IIx - all = e
then
P(x) = 1I(I(x)-f(a))-(Y-f(a))11 2 > (ce-(1/2)C£)2
= (1/4)c 2 £2 > lIy - f(a)1I 2 = P(a),
so P must assume its minimum value at an interior point b, lib - all < £. Then
o = (8P/8xj)(b) = L 2 (Ii (b) - Yi) . (8fi/EJXj)(b) for all j,
that is, dbf(l(b) - y) = O. Since dbf is invertible this gives f(b) = y, and we
have proved the claim. Since e can be taken arbitrarily small this argument shows
that f maps each neighborhood of a onto a neighborhood of f(a). Put U' :=
{x: Ilx - all < £}. For the same reason f maps each neighborhood of each point
xE U' onto a neighborhood of f(x). Since f is also injective on U' it follows that
f maps U' homeomorphically onto the open set feU'). It remains to show that
f- I : f( U') -+ u' is a CI- map .
Let yEf(U') approach f(a) and put f-I(y):= bEU', so b approaches a. From
(I(b) - f(a) - daf(b - a))/Ilb - all-+O as b-+a,
and lib - all :5 c-Illy - f(a)li we obtain by applying (da!)-I
((d a ffl(y - f(a)) - (I-l(y) - a))/lly - f(a)II-+O as y-+f(a).
Hence f- I is differentiable at f(a) with d J Ca)f- 1 = (d a !)-I. For the same reasons
f-l is differentiable at every point y E f( U') with d y f-l = (d J-ICy)f) -1. Thus f-l
is CI. 0
COROLLARY (IMPLICIT FUNCTION THEOREM). Let U  Rm+n be a definable open
set and fl"'" fn : U -+ R definable CI-functions. Let (xo, Yo) E U be such that
fl (xo, Yo) = . . . = fn(xo, Yo) = 0 and the n X n matrix ( EJ fj/8Yk)(XO, Yo)) 1<' k<n
is invertible. Then there are open definable neighborhoods V of Xo in Rm a-d w
of Yo in R n respectively, and there is a definable CI- map <f> : V -+ W such that
V x W  U, rf;(xo) = Yo, and such that for all (x, y) E V X W we have
!t(x,y)='.'=fn(x,y)=O {:io y=rf;(x).

114
SMOOTHNESS
PROOF. Apply the inverse function theorem to the map
(x,y) 1-+ (x,f1(x,y),.. .,/"(x,y»):U_R m +,,. 0
(2.12) EXERCISES.
1. (L 'Hapital's rule) Let I be an interval and I, g: I - R definable functions, and
let a be one of the endpoints of the interval, possibly a = +00 or a = -00. SUPI.)()SP
that g'(x) I- 0 for all x E I in some neighborhood of a, and that limx....al(x) =:
Hmz--oag(x) = 0, or limz--oa If(x)1 = limx....a Ig(x)/ = +00. Then
lim(J(x)/g(x») = lim(J'(x)/g'(x»).
z--oa z--oa
(Note that both limits exist in Roo, by Chapter 3, (1.6).)
2. (Taylor's formula) Suppose the definable function I: I - R is (n + 1) times
differentiable on the interval I, and let a, bE I, a < b. Then
I(b) =
I (Z) ( a ) N") ( a )
I(a) + J'(a)(b - a) + -(b - a)Z + .,. + L2--(b - a)"
2! n!
1 ("+1) ( )
+ Z (b _ a)"+l
(n + 1)!
for some z with a < z < b.
93. Definable maps are piecewise C 1
(3.1) Another basic tool is a Cl- vers ion of cell decomposition. First we extend t hI'
notion of Cl- map and define Cl- ce ll s .
DEFI:I'ITION.
(1) A definable map I:A_Rn, where A S;; Rm is not necessarily open, is said
to be a Cl. map if there are a definable open set U S;; Rm containing A and
a definable Cl- map F: U..... R n such that PIA = I. (Then I is continuon"
and for open A this gives the usual notion of Cl- map on an open set.)
(2) The notion of C 1 -cell is defined inductively as in Chapter 3, (2.3), except
that when forming r(J) and (/,g) we now require the functions I and lJ
(when R-valued) to be Cl (and definable) instead of just continuous (and
definable).
Note that every inclusion map A..... R m (for definable A  Rm) is Cl, and that if
I:A_Rn is C 1 and 9 :B.....RP is C 1 , with B  R", then go f :f-l(B)_RP is C 1
Also I = (/1, . . ., In): A - Rn is Cl iff each function fi: A....,. R is Cl.

SMOOTHNESS
115
(3.2) THEOREM (C 1 -CELL DECOMPOSITION).
(1m) For any definable sets AI,.. ., Ak  R m there is a decomposition of Rm into
C 1 -cells partitioning AI, . . ., Ak'
(11m) For every definable function f: A  R, A  Rm, there is a decomposition
of Rm into Cl-cells, partitioning A, such that each restriction flC: C  R
is Cl for each cell C  A of the decomposition.
If f and A are as in (I1m) and p is an interior point of A, we define
v f(p) ;== ((af/8xt)(p),..., (8f!8x m )(p»),
provided these partials exist at p. If some partial is not defined at p, then v f is
not defined at p. Further we put
A' := {PE A: p is an interior point of A at which v f is defined}.
Alon!!; with (1m) and (I1m) we will prove the following technical result.
(IlI m ) A - A' has empty interior.
PROOF OF THE THEOREM. By induction on m: (Id is trivial and (1II 1 ) is a re-
formulation of (2.5). To prove (Ill)' let f: A  R, with A  R, be definable. By
(2.5) there is a decomposition 1) of R partitioning A such that the restriction of
f to each interval in 1) contained in A is differentiable, and by the monotonicity
theorem we may assume, after refining 1) if necessary, that the restriction of f to
each interval in 1) has continuous derivative.
Kow we shall assume inductively that (Id), (lId) and (lII d ) hold for all d:S m, and
derive successively (Im+d, (III m +d and (IIm+d.
PROOF OF (Im+l)' Let AI,,'" Ak  Rm+1 be definable. We want to find a Cl_
decomposition of Rm+1 partitioning AI",', Ak. By ordinary cell decomposition
there is a decomposition 1) of Rm+l partitioning AI,..., Ak. Then 7r(1) is a
decomposition of R m where 7r : Rm+l  Rm is the usual projection map. Let
1!'(V) = {Cl, . . ., Cn}, and for each i = 1,. . ., n let the cells of 1) that project onto
C i be
(-OO,f.l)' r(/.I), (/.1.f.2),..., r(/i.), (/i..+ OO ),
where f'b"', fi. ; C i  R are definable and continuous, s = s(i). By (1m) and
(U m ) we may assume, after suitably refinin!!; 7r(1), and 1) accordingly, that all C i
and all fij are ct. Then 1) is a Cl-decomposition as required. 0
PROOF OF (III m + 1 ). Let f: A  R be definable, A  Rrn+l, and define A' as
above. Consider an open box U x (a, b)  A. It suffices to show that U x (a,b)
intersects A'. By (2.5) and definable choice we can pick for each pE U an interval

116
SMOOTHNESS
(CI'(p),,6(p))  (a,b) such that CI' and {j are definable R-valued functions on Valld
81/8x m +1 is defined on {p} x (a(p),,B(p)). Using cell decomposition we can shrink
V so that CI' and ,6 are continuous on V; shrinking V further and changing a alld
b we may as well assume that a f / aXm+l is defined on the entire box V x (a, b).
Take any t E (a, b). By applying (IIIm) to the function p ....... I(p, t) : V -+ R we se!'
that there must exist Po E U such that all partials (a J/ aXi)(PO, t) for i = 1,. .., 1Ii
are defined. Hence (Po,t)EA', as desired. 0
PROOF OF (IIm+d. Let I: A -+ R be definable, A  Rm+l. Let A' be as above.
Take a decomposition V of Rm+l partitioning A and A' such that V' f (and hene('
J) is continuous on each open cell of V contained in A'.
CLAIM. For each cell C E V with C  A, dim(C) < m + 1, there is a decomposition
V c of R m +1 partitioning C such that liD is C 1 for each DE Vc.
PROOF OF CLAIM. Let CEV, C  A, dim(C) = d $ m, and consider the homeo
morphism Pc: C -+ p( C) onto the open cell p( C)  Rd. By (lId) we can partition
p( C) into finitely many cells B such that 10 Pc/ IB is C 1 .
By composing with the Cl.map Pc we obtain that IIPc1(B) is C 1 . Take for 'Dc a
decomposition that partitions each of the sets Pc/(B). This proves the claim. 0
Now we are done: by (Im+d there is a decomposition V' that refines V and all Tic
as in the claim, and that consists moreover entirely of C1-cells. Let C' E V' and
C'  A. It suffices to show that flC' is Cl. Take the cell C E V that contains C ' ,
so C  A. If C is open, then by (IIIm+d the cell C intersects A', hence C  A'.
so flC is c 1 , so IIC' is Cl. If C E V is not open, apply the claim to conclude that
IIC'isC 1 . 0
(3.3) EXERCISES.
Let f = (/1,..., fn): U -+ R n be a definable map on a (definable) open set U  R m
We define inductively what it means for f to be a Ck- map , where k is a positiv!'
integer. For k = 1 this has been defined in (2.6). For k > 1 the map I is said to lw
a Ck-map if I is a Cl-map and df: V -+ Rnm is a Ck-l. map .
Next, let I: A -+ Rn be a definable map where A  R m is not necessarily open.
Then we define f to be a Ck-map by replacing in (3.1) everywhere "C 1 " by "C k ".
Similarly, define Ck-cells as in (3.1) by replacing everywhere "C 1 " by "C k ".
1. Show that the remarks at the end of (3.1) go through with "Cl" replaced by
"Ck".
2. State and prove the Ck-Cell Decomposition Theorem, k  1.

SMOOTHNESS
117
54. Existence of good directions
(4.1) For each xERm with Ilxll < 1, put
vex) :== (xl,...,xm, V l-lIxI12),
so that Ilv(x)1I = 1, v(x)ER m + 1 . (Kote: vex) is the point on the unit sphere sm 
Rm+l lying directly above x.) Let A  Rm+l be definable with dim(A) < m + 1.
We now prove the final result of this chapter, the existence of "good directions" for
A. It will be crucial in triangulating definable sets in the next chapter.
(4.2) THEOREM (GOOD DIRECTIONS LEMMA). Let B  Rm be a box contained in
the disc II xl; < 1. Then there is x E B such that for each point p E Rm+l the set
{tER: p+t'V(X)EA} is finite.
For such x, every line in the affine space RmH with direction vex) intersects A in
only finitely many points. We then call v(x) a good direction. So the theorem tells
us that the set of good directions is dense in the unit sphere sm  Rm+l.
PROOF. Suppose there is no such x. Then for each x E B there is PE A and f > 0
such that p + t. vex) E A for all t with It I < f. By definable choice there is then a
definable map assigning to each x E B a pair (p, f) with these properties. Applying
(3.2) we may even assume, after shrinking B, that there are a C 1 -map P:B-+RmH
and a fixed f > 0 such that P(x) + t. v(x) E A, for all x E Band It I < f. Define the
Cl- map 4>: B x R-+ Rm+l by: <I>(x, t) :== P(x) + t . vex). Let us fix a point x E B.
CLAIM. There is t, Iti < f, such that the R-linear map d(z;,t)4>: RmH - Rm+l is
invertible.
To prove the claim we consider the matrix M(t) of d(z;,t)<I> with respect to the
standard basis el, . . . , em, emH. (In this context the elements of Rm+l are column
vectors written as (al,"', a m +l)l, the "t" standing for "transpose".) Let P =
(Pl'''', Pm+d and put Pi := ((8P1/8xi)(X),.", (8Pm+d8xi)(X))' for 1 :$ i :$ m.
A straightforward computation shows that the columns of M(t) are given by
M(t) . Ci
M(t) . em+!
== Pi + t . (Cj - (xd V I - IIx 11 2 ) . em+1) , 1 :$ i :$ m,
vex).
It follows that det (M(t») is a polynomial in t. For t I- 0 the m + 1 columns of M(t)
are independent if and only if the m + 1 vectors
(pdt) + ej - (xi/ V I -lIxI12) . em+! (i = 1,. ", m), and vex)

118
SMOOTHESS
are independent. When t is large these m + 1 vectors are close to the m + 1 vec()r
fi - (x;/ V 1-llxI/2) . em+! (i = 1,...,m), and vex),
which are easily seen to be independent. So for sufficiently large t the m + 1 cOlunlll'
of M(t) are independent. Therefore the polynomial det (M(t») is not idelLLira:y
zero. So det(M(t») 1 0 for some t, It I < (, which proves the claim.
Take such a t, It I < €, with dc.x,t)</> invertible. From (2.11) it follows that ,\ is
injective on a box Bo around (x, t). Take Bo so small that Eo <:;; B x (-€, f). Th('11
if>(Bo)  A by definition of </>. But dim(if>(Bo)) = dim(Bo) = m + 1, contradiclill!1,
dim(A) < m + 1. 0
(4.3) EXERCISES.
1. Let A  Rm+l be a definable set of dimension :$ m. Call a unit vector 1/ E
sm  Rm+l an asymptotic direction for A if for each ( > 0 and r > 0, (JI('1'(' is
a point x E A with I/xl/ > r and II (x/l/xl/) - ull < (. Show that the (definable) '('
of asymptotic directions for A is of dimension < m. (In particular, not every unit
vector is an asymptotic direction for A.)
2. Let A  R m be definable and dim(A) :$ k < m. Show that there is an (m - A')-
dimensional linear subspace L of Rm all of whose translates v + L (v E Rm) nw('(
A in only finitely many points. (Hint: proceed by induction on m - k.)
3. In this exercise, assume that R is an ordered field, not necessarily real closed.
Let A  R m be semilinear with dim(A) < m. Show that A is contained in a finih'
union of affine subspaces of Rm of dimension < m, and derive that there is a !!.ood
direction for A, that is, a nonzero vector u E R m such that each line p+ Ru (fiE f{"')
intersects A in only finitely many points.
Notes and comments
Proposition (2.5) that definable functions are piecewise differentiable is adapt<'d
from the appendix in [19;, which also shows C 1 -ccll decomposition in the das,;ical
real case assuming "strong o-minimality". The proof ofthe inverse function theoI'<'nl
is along the lines of that for polynomial maps in Brumfiel [7]. The good directioJls
lemma (4.2) was inspired by a lemma due to Koopman and Brown [36], which also
appeared in (unpublished?) lecture notes by Sussmann on subanalytic sets (:::id
1980s ).

CHAPTER 8
TRIANGULATION
Introd uct ion
We work jn this chapter wjth a fixed o-mjnimal expansjon (R,<,S) of an ordered
real closed field (R, <,0,1, +, -, .).
We point out, however, that most of Section 1 is of a purely semilinear character,
and makes sense over any ordered field. In Section 2 we prove the main result in
this chapter, the triangulation theorem, which allows us to reduce many questions
to the semilinear case. In Section 3 we use triangulation in this way to show that a
definable continuous function f: A...... R on a definable closed subset A of a definable
set B can be continuously and definably extended to B. This result is needed in
the next two chapters on trivialization and quotient spaces.
1. Simplexes and complexes
For simplicity, we denote the ordered field (R,<,O,I,+,-,.)just by R. In this
section the role of R is mainly that of an ordered vector space over itself. In
particular we will only deal with subsets of R n that are semilinear over R, in the
sense of Chapter 1, (7.9). We call a map f: A...... R n with A  R m semilinear if
its graph ru)  Rm+n is semilinear.
(1.1) An affine subspace of R n of dimension d is by definition a translate L + a of
a linear subspace L of Rn of dimension d. For d = 1 this is also called a line, and
for d = 2 we speak of a plane. Given points ao, al, . . . , ak in R n we have
aO+R(al-aO)+...+R(ak-ao) ={),a;: tiER, L:t;=l}
= the smallest affine subspace containing ao,..., ak.
This affine subspace is called the affine span of ao, . . ., ak.
(1.2) A tuple ao,.. ., ak of points in R n is called affine independent if the affine
span of ao,. . ., ak has dimension k. For k = 0 this is always the case, for k = 1 it
lIleans that ao '" al, for k = 2 that ao, aL, a2 are distinct and not on a line.
Affine independence of ao, al, . . . , ak is equivalent to the condition that the k vectors
al - ao, . . ., ak - ao are linearly independent.
119

120
TRIANGULATION
Let ao,..., ak be affine independent, aj ERn. Then the map (to,.. ., tk) H L: tilli
is a homeomorphism from the affine subspace {(to,...,tk)ER k +1: L:tj = I} of
R k +1 onto the affine span of ao,..., ak. In particular, each point x in the aflill p
span of ao,. .., ak has a unique representation as
x=toao+...+tkak with L:tj=l,
and to,. .., tk are called the affine coordinates of the point x with respect to
ao,..., ak. (Sometimes also called barycentric coordinates.)
(1.3) A set S  R fi is called convex if for every two distinct points a, bin S tlip
line segment [a,b] := {ta + (1- t)b: 0:::; t :::; I} is contained in S. The convex
hull of a set S  Rn is the smallest convex subset of Rfi that contains S, and is
easily seen to equal the set of all points L: tja. with ao,. . ., ak E Sand L: t. = '.
t.  O.
Given a convex set C in R fi , a point p E C is called an extreme point of C if for
all Pl,P2 EC such that p = (Pl + P2)/2 we have P = Pl =]}2.
(1.4) DEFINITIONS. An affine independent tuple of points aO,al,...,ak E R" is
said to span the simplex
(ao,...,ak) :={L:t.a i : allt.>O, L:t.=l}Rn.
We call (ao,..., ak) a k-simplex in R n . Note that it is open in the affine spall of
ao, . . ., ak. Its dimension as a semilinear set is easily seen to be k.
The (topological) closure of (ao, .. .,ak) in Rfi is denoted by [ao,.. .,ak], so that
[ao,.. .,ak] : = {L:tja.: all t. 2: 0, L: tj = I}
= the convex hull of {ao,.. .,ak}.
One easily checks that ao,. . ., ak are exactly the extreme points of [ao,..., Uk]' W,-
call ao,.. .,ak the vertices of(ao,.. .,ak) and also of[ao,.. .,ak]' Instead of "S is
the set of vertices of the simplex (1" we also say "S spans (1".
A face of (ao, . . ., ak) is a simplex spanned by a nonempty subset of {ao, . . .,1/ d.
Note that distinct nonempty subsets of {ao,..., ak} span disjoint faces, and t Ita:
ao,.. .,ak] is the (disjoint) union of the faces of(ao,.. .,ak).
Given simplexes (1 and T we write (1 < T if (1 is a proper face of T, that is, (1 is a
face of T and (11 1'.
The barycenter b«(1) of a k-simplex (1 = (ao,..., ak) is the point
1
b«(1) = -(aD +... + ak),
k+l

TRIANGULATION
121
which belongs to (1.
EXAMPU;S OF CLOSUR;S OF SIMPLEXES.
[au]
lao, a1]
(ao) :;:: {au}; however, aj  (ao,..., ak) iU > 0;
(ao, ad U {ao} U {ad, the line segment between
distinct points ao and al;
:;:: the "triangle" spanned by distinct points ao, ai, a2 not on a line;
the "tetrahedron" spanned by four distinct points ao, ai, a2, a3
that do not all lie on one plane.
[ao, a!, a2]
LaO,al,aZ,as]
(1.5) DEFINITIONS. A complex in R" is a finite collection K of simplexes in R",
such that for all (11, (12 E K,
either cI( (1d n cI( (12) :;:: 0,
or cI«(1J) n cI«(12) = cI(T)
for some common face T of (11 and (12. (This T is not required to belong to K, which
makes our definition somewhat more general than is usual. The usual definition of
"complex" corresponds to what we call "closed complex" below.)
l\'ote that distinct simplexes in K are disjoint. We put
IKI .- union of the simplexes of K, a bounded semilineaf subset of R";
Vert(K) .- the set of vertices of the simplexes in K, a finite subset of R".
We also call1KI the polyhedron spanned by the complex K. Note: the notational
conflict that iKI also denotes the number of simplexes of K is always resolved in
favor of "IKI = polyhedron spanned by K". (All bounded semilinear sets in R"
are polyhedrons, that is, of the form IKI for a suitable complex K in R", see (2.14),
exercise 2.)
A subset of a compJex K in R" is also a complex in R", and is called a subcomplex
of K.
A complex K is called closed if it contains with each simplex all its faces. Equiv-
alently, a closed complex in R" is a finite collection K of disjoint simplexes in R"
snch that each face of a simplex in K also belongs to K. Note also that a complex
l( in R" is closed iff IKI is closed in R".
ven a complex Kin R" we put K := the set of faces of the simplexes in K. So
II: is clearly a closed complex in R", with K as a subcomplex, and
cllKI = IKI.

122
TRIANGULATION
Let L be a sub complex of K. Then ILl is closed in IKI if and only if L n K ::: L.
In that case we also say that L is closed in K.
Note that a complex K in R" is completely determined by the finite set Vert (1\ ) C;;
R n and by the collection of subsets of Vert(K) that span simplexes in K.
(1.6) Let J( and L be complexes in R m and R" respectively.
Then a vertex map V:K  L is a map V :Vert(K)..... Vert(L) such that whell('\PI'
(ao,...,ak) is a k-simplex of K, then {V(ao),...,V(ak)} spans a simplex of [,.
(We allow V(a;) ::: V(aj) for i :F j.)
Such a vertex map V:K L determines a continuous 'map !VI; IKI  ILl by:
!VI (L:t;ai) = L:tiV(a;)
where (ao,.. ., ak) is a k-simplex of K, ti > 0, L: t; = 1.
To show that !VI is continuous, show first that the restriction of !VI to the clOSi::.('
of (ao, . . ., ak) in IKI is continuous. Then use the fact that if f: X  Y is a ""if>
between topological spaces X and Y, and X = Xl U... U Xk a covering of X by
finitely many closed sets X; such that each restriction fiX; :X;..... Yi is continuous,
then f is continuous. This general fact will be used frequently.
 ote also that the map !VI is semilinear, and that if W : L  M is a second v('rl ex
map, then the composition W 0 V: K ..... M is a vertex map such that
IWoVI = IWlo!Vl.
We now state our main result concerning trian!!;ulation.
(1.7) TRIANGULATION THWR;M. Each definable set S  Rm is definably howl!-
morphic to a polyhedron IKI for some complex K in Rm.
This theorem expresses the truly remarkable fact that the (definable) topoloy of
a definable set can be completely described in finite combinatorial terms. To s('e
this, let K be a complex, and consider its scheme
(Vert(K), {S  Vert(K); S spans a simplex of K}) ,
which is just a finite set equipped with a collection of subsets, a finite combinato;ial
object par excellence. For L to be a complex with isomorphic scheme means LJ,('I"('
is a bijective vertex map V:K  L whose inverse is also a vertex map. Such a iliaI'
V induces a semilinear homeomorphism !VI: IKI  ILl. In this sense, the schE'I!I{' of
K, as a finite combinatorial object, determines the topology of the polyhedron!" I
up to semilinear homeomorphism.
We prove this theorem in a more precise version in the next section. In the rest of
this section we consider some elementary operations on complexes.

TRIANGULATION
12:1
(1.8) BARYCENTRIC SUBDIVISION. Let K be a complex. A K -flag is a sequence
: 0'0 < 0'1 < . .. < 0' f (J  0)
of simplexes, with O'f E K and each 0', a proper face of the O'j with j > i.
We do not require 0', E K for i < f, or that 0', is of dimension i.
To such a K-flag  we associate an f-simplex b() := (b(O'o),...,b(O'f») whose
vertices are the barycenters of the simplexes of;Y. One easily checks that b() is
indeed an f-simplex, that b()  O'f' and that if ;Y1 and 2 are distinct K-flags
then b(;Y1) and b(2) are disjoint.
The (first) barycentric subdivision of K is the complex K' whose simplexes are
the simplexes b() associated to the K-fiags. One checks easily that the b()
indeed form a complex, and that IK'I = iKI.
(1.9) We need one further general construction.
Let (ao,.. .,ak) be a k-simplex in R", and r"s, E R, r, $ s, for i = 0,.. .,k, and
rj < Sj for some j. Put b, := (ai,r;), C, ;= (a"s,) E R"+l. Then (bo,...,bk),
(co,..., Ck) are k-simplexes in R,,+1. One also easily checks that if b, 1 c, (that is,
Ti < s,) then (bo,..., bi, Ci,"', Ck) is a (k+ I)-simplex. We now construct a complex
L in R"+1 such that ILl is the set of points "between [bo,..., b k ] and [co,.. ., Ck]".
(1.10) LEMMA. Let L consist of all (k + I)-simplexes (b o ,.. .,b;,c".. .,Ck) with
h, f- c" and all faces of these (k + 1 )-simplexes. Then L is a closed complex and
ILl = {t(tob o +... + tkbk) + (1 - t)(toco +... + tkCk) :
O::;t$l, t,O, L:t,=I}
convex hull of {b o ,..., bk, Co,..., Ck}'
PROOF. The proof of this lemma is rather long, and is divided into four parts.
PART I. In this part we show L js a closed complex. Let 0' and T be faces of
(b o ,..., bi, Ci,"', Ck) and (b o ,..., hi> Cj,..., Ck) respectively, ri < 5i, Tj < Sj.
Suppose 0' n T 1 0. We have to show that 0' = T. This is clear if i = j since then 0'
and T are faces of the same simplex. So assume i < j, and let x EO' n T. Write
x == tob o +... + tibi + tCi + t'+lC'+1 + ... + tkck, with t + to +... + tk == 1,
x = uoho + . . . + Uj-1 b j - 1 + ubj + UjCj + . . . + UkCk, with U + Uo + '" + Uk = 1.

124
TRIANGULATION
Applying the projection map ?I': RnH  R n to these expressions for z and consid.
ering the affine coordinates of ?I'(x) with respect to ao,.. ., ak we get
(*) ti+t = Ui, ti = uj+u,andt.\ = u.\forAE{O,...,k}, A1i, A1j.
Also the (n + 1 )Ih coordinate of x satisfies
Xn+1 = toTo + ... + tiTj + tSi + t j HS;+1 + ... + tksk
= UOTO +... + Ui-1Tj-1 + UTj + UjSj +... + ukSk.
Subtracting tOTO + ... + tj-l Tj_1 + tj+1Sj+l + .,. + tksk from both sums yields
tjTi + ts; + L t.\s.\ =
;+l.\j
L U.\T.\ + UTj + UjSi'
i.\:Sj-1
Subtracting the right side from the left and using (*) we get
t(Si - T;) + L t.\(s.\ - T.\) + U(Sj - Tj) = 0,
;+l.\i-l
in which each term on the left is nonnegative. Hence each term equals zero, so
t = u = 0, and t.x = 0 whenever i + 1  A  j -1 and T.\ < s.\. The vertices of a are
exactly the points among bo,. . ., b j , Cj, . . ., Ck with respect to which x has a naill-pro
affine coordinate, and similarly with T and bo,.. ., bi, cj,. . ., Ck. This shows that a
and T have the same vertices, so that a = T.
PAR.T II. In this part we prove that ILl contains the set
{ t (tobo + .. . + tkbk) + (1 - t)(toco + . .. + tkCk) : 0  t  1, t;  0, L tj :- I} .
Let x = t(tob o +" .+tkbk) +(I-t)(toco +.. .+tkCk), 0  t  l,t;  0, :E Ii :- l.
We have to show that x E ILl. Clearly the (n + 1)lh coordinate XnH of z satisfies
toTO + ... + tkTk  ZnH  toso + ., . + tksk,
so there is an iE {O,..., k} such that T; < S; and
toTo + .,. + tj-1 Tj-1 + tjTj + t;+1sH1 + ... + tksk  xn+l
 toTo + . .. + tj-l Tj-1 + tjSj + tH1S;+1 ., ... + tkSk.
Hence X n +1 = toTo +... +ti-l T;-1 + U + tHI Si+l +... + tksk, with t;T;  a -:; li"i.
Then we can write U = UTi + VSi with U, v  0 and U + v = ti. Thus
X n +1 = toTo + ... + tj-l Tj-l + UTi + VSi + tH1Sj+1 + . . . + tkSk'
Also
1I'(z) = L tjU; = toao + . . . + ti-la;-1 + UUj + vaj + t.+l a i+1 + . .. + tkUk'

TRIANGULATION
125
Therefore
x = tobo + ... + ti-lbi-l + Ubi + VCi + ti+1Ci+l + ... + t"ck,
since the right hand side has the same (n + 1 )Ib coordinate and the same projection
in R n as x. This expression for x shows that x E [bo, . . ., b;, Cj, . . ., Ck], and finishes
the proof of part II.
PART III. The set 5:= {t:Etibi + (1- t) :EtiCj: 0  t  1, ti  0, :Et i = I} is
convex.
To prove this, first a preliminary remark.
For a point xE 5, x = t:Etibi + (1- t):Etici with 0  t  1, ti  0, :Eti = 1,
we have 7I"(x) = :E tiai E [ao,..., ak]. Hence, given a point p = :E tiai E [ao,..., ak],
ti  0, :Eti = 1, the vertical line 7I"-l{p} = (p,O) + He n +1 intersects 5 exactly in
the line segment [:E tibi, :E tiCi] , and the set of (n + 1 )tb coordinates of the points
on this line segment equals [:E tiTi,:E tiSi]'
Let now x, y E 5, and consider a convex combination z = ux + vy, u, v  0,
u + v = 1. We have to show that z E S. Write x = t:E tibi + (1 - t) :E tici as above,
and similarly y = t':E t:b i + (1 - t'):E *i. Then
7I"(Z) = U7I"(x) + v7l"(Y) = L (uti + vtDai E [ao,..., ak],
and the (n + 1)lb coordinates satisfy
Zn+l = uXn+! + vYn+1, L tiTi  x n +1  L tis;
and :Et:Tj  Yn+1  :Et:Si' Hence :E(uti + VtDT, ::; zn+1 ::; :E(ut, + vtDsi. so
that by the preliminary remark we have zE S.
PART IV. Rest of the proof. The set S from part III obviously contains the points
b o , . . ., bk, Co, . . ., CIt, hence it contains the convex hull of these points. Finally, the
convex hull of these points contains each set [b o ,..., b i , Ci,"., CIt] with b j 'I C"
hence it contains the union I LI of these sets. This completes the circle of inclusions,
and thereby the proof of the lemma. 0
(1.11) REMARKS. One should note that the complex L of our lemma depends on
the listing of the vertices of (ao, . . . , a k) in the order indicated: first ao, then aI,
etc. On the other hand, its polyhedron ILl is independent of this ordering. Note
also that the simplexes (b o , . . ., b k ) and (co, . . ., Ck) are in L, and that 71"( (1) is a face
of (ao, . . ., ak) for each C1 E L.

126
TRIANGL'LATION
(1.12) EXERCISES.
To state the first exercise, define a map I: Rm -> R" to be affine if it is of th0 tornl
I(x) = L(x) + a where L: R m -> R" is linear, and aE R n . Note that for n == 1 t hi,
is just what we called an affine function in Chapter 1, Section 7.
1. Show that the composition of affine maps is affine, and that if I: Rm -+ II" j,
affine, then I('L tjai) = 'L t;/(a;) for aj E R m , and tj E R with 'L ti = 1. Conclnd"
that if two affine maps R m -+ Rn take the same value at points ao,. .., ak E N".
they agree on the affine subspace of R m spanned by ao,..., Uk.
2. Let T : Rm -+ Rm be an R-linear automorphism, or a translation, and !PI
ao,. . ., ak be an affine independent tuple of points in Rm. Show that the t npl"
T(ao),..., T(ak) is affine independent and that
T((ao,...,ak)) = (T(ao),...,T(ak)).
3. Let (1 be a k-simplex in R m . Show that there are m-k polynomials It, . . ., fm- ,E
R[Xt,.. .,X m ] and k + 1 polynomials 90,.. .,9kER[X1,.. .,X m ], with all j's and
9'S of degree 1, such that
(1 = {xER m : fj(x) = 0, 9j(X) > 0, 1 $ i $ m - k, 0 $ j $ k},
and
cl((1) = {xER m : I;(x) = 0, 9j(X)0, l$i$m-k, O$j$k}.
Hint: let the affine space spanned by the vertices of (1 be given by the systelll of
equations J;(x) = 0 (1 $ i $ m - k).
4. Let V : K -+ L be a vertex map between complexes K and L, and let IT E 1\.
Show that iVl((1) is the simplex of L spanned by {V(p): p is a vertex of (1}.
5. Let K be a complex in R m , (1EK and dim((1) = dim(IKI). Show that thpn IT j,
open in IK:.
6. (CaratModory) Show that the convex hull of a set S  R m is the unioJl of the
convex hulls of its finite subsets of size $ m + 1. (Hint: assume 181 = m T 2.)
7. Show that the convex hull of a definable set in Rm is definable.
(1.13) REMARK. Everythinl!; in this section (except (1.7) and exercise 7 a},{)\P:
makes sense and is valid for an arbitrary ordered field R, not necessarily real cloSt,d.

TRIANGULATION
127
2. Triangulation theorem
first some facts on extending continuous definable functions.
(2.1) L;MMA. Let (1 be a k-simplex in R m , k > 0, and f:{)(1--+ R a continuous
definable function. Then f has a continuous definable exten8ion g: cI( (1) - R.
PROOF. Let b = b((1) be the barycenter of (1. Connect each point pE{)(1 to b by the
line segment [P,b] = {tp+ (1- t)b: 0  t  I}, and define 9 on this line segment
by g(tp + (1 - t)b) = tf(p). This gives 9 the value 0 at the barycenter b, and 9
is well defined on all of cI( (1) since the line segments cover .cI( (1) and line segments
corresponding to different p's only intersect at the point b. Moreover 9 extends f.
By viewing cI((1) as the continuous image of the map
(t,p)...... tp + (1- t)b : [0,1] x (cI«(1) - (1) _ R m
we can appeal to Chapter 6, (1.13) to conclude that 9 is continuous. 0
(2.2) LEMMA. Let K be a closed complex in Rm and L a closed subcomplex. Then
each continuous definable function f: ILI--+ R has a continous definable extension
j:IKI-R.
PROOF. It will suffice to show that if L 1 K, then we can find a strictly larger closed
subcomplex L ' of K and a continuous definable extension f' : IL'I - R. Assuming
L 1 K, take a simplex (1 E K - L of minimal dimension. If (1 is a O-simplex, then
11 = {a} for a point a rt ILl, L U {(1} is a strictly larger closed subcomplex of K,
and 1 has continuous definable extension f': IL U {(1}1 = ILl U {a} - R given by
f'(a) = O. Next assume (1 is a k-simplex with k > O. Then all proper faces of (1
arc in L, so cl«(1) - (1  ILl, and by the previous lemma the function fl cI«(1) - (1
extends continuously to a definable function g: cI( (1) _ R. Further L' := L U {(1}
is a strictly larger closed sub complex, and f extends continuously to the function
I':/L'I = ILl u cl«(1)-R defined by
I'(x) = f(x) for xEILj,
I'(x) = g(x) for x EcI«(1). 0
(2.3) D;f'INITIO:-;. Let A  Rm be a definable set. A triangulation in R n of A
is a pair (4), K) consisting of a complex K in R" and a definable homeomorphism
cJ>:A_IKI. Note that then
4>-l(K) := {4>-1«(1): (1EK}
is a finite partition of A. We call (A,4>-l(K)} a triangulated set. The trian-
gulation is said to be compatible with the subset A'  A if A' is a union of
elements of cJ>-l(K). Note that by Chapter 6. (1.10), we have
A ill closed and bounded {:} the complex K is closed.

128
TRIANGULATION
In that case each continuous definable R-valued function on cl(G), where C E
c}-1 (K), has a continuous definable R-valued extension to A, by lemma (2.2).
(2.4) Let (A, P) be a triangulated set, and C,DEP.
We call D a face of G if D  cl(C) and a proper face of C if D  cl (C) - C.
Note that if P = ;P-l(K) for the triangulation (;P, K) of A, then D is a (prop('r)
face of C iff the simplex ;P(D) is a (proper) face of the simplex ;P(C). Hence,
If D is a proper face of C and y E D, then there arc arbitrarily small definaIJ/r,
eighborhoods U of y in A such that U n C is definably connected. (To ,see t I:is,
use the fact that a convex definable set is definably connected.)
Note also that cl( C) n A = union of the faces of C in P, for G E P.
(2.5) DEFINITION. A multivalued function F on the triangulated set (A, 7')
is a finite collection of functions, F = {fe,.: C E P, 1  i  k( C)}, k( G) > 0,
each !c,; :G---> R continuous and definable, and !c,l < ... < fC,k(C)' We set
r(F) .- U r(J),
JEF
FIG .- {fe,;: 1  i  k(G)}for GEP,
pF ._ {r(J): fEF} U {(Je,;,Jc,i+d: Cep, 1  i < k(C)}.
A F .- the union of the sets in pF,
so pF is a partition of AF  Rrn+l and 'II'(r(F») = 'II'(AF)  A. Here '11'; Rm+l --
Rrn is the projection map onto the first m coordinates.
We consider two properties of a multivalued function F on (A, P):
(1) We call F closed if for each pair C, D E P with D a proper face of G and each
f E FIG there is g E FiD such that g(y) = lim..-y f( x) for all y ED.
Note that then each f E F, say f E FIC, extends continuously to a definable functio:l
clU): cl( C) n A ---> R, that the restrictions of cl(J) to the faces of C in P belong to
F, and that r(F) is closed in A x R. (Lemma (2.6) below is a converse to this.)
(2) We call F full if F is closed, k(C)  1 for all Cep, and for each pair C,DCr
with D a proper face of C and each g E FID we have g = cl(J)JD for some f E F, C,
where cl(J) is the continuous extension of f to cl(G) n A.
Note that then 1t'r(F) = A, and that if A is closed and bounded, so is AF.
(2.6) LEMMA. Let F be a multivalued function on the triangulated set (A, P) 8U"/'
that f(F) is closed in A x R, and there is M > 0 such that r(p)  A x [-M, A(.
Then F is closed.

TRIANGULATION
129
paOOF. Let C, D E P, D a proper face of C, and let IE FIC. Given yE D we have
to show first that lim:l>-'\l f( x) exists. Set I := lim inf:l>-'\l f( x), L := lim sUP:l>-'\l f(z)j
Le. I = inf { t E R : there are x E C arbitrarily close to y with f( x) < t}, and similarly
for L. Clearly -M :s; l:s; L:S; +M, and 1= L implies that lim:l>-'\lf(x) exists and
equals l. Suppose I < Lj take any t with I < t < L. By the remark in (2.4) there are
arbitrarily small definable neighborhoods U of y in A such that un C is definably
connected. Since f takes both values < t and values > t on U n C it takes also the
value t on Un C. Hence (y,t) EcI(r(J»)  reF) for each t in the interval (l,L).
But there can only be finitely many t with (y, t) E reF). This contradiction shows
1= L = lim:l>-'\lf(x). Set I/I(y) = lim:l>-'\lf(x). One checks easily that q;:D....R is
continuous, and since r( F) is closed it follows that <f>E F. 0
(2.7) MORE TERMINOLOGY. Let (4),K) be a triangulation in R" of A  R m , and
let B  Rm+1 be a definable set, B  A X R. Then a trian!!;ulation (ili, L) in R"+1
of B is said to be a lifting of (4), K) if K = {'II'( (1): (1 E L} and the diagram
B .
----;
1
A ----;
.
ILl
1
IKi
commutes where the vertical maps are restrictions of the projection maps Rm+1 ....
Rm onto the first m coordinates and R"+1 .... R" onto the first n coordinates. (In
particular, B projects onto A.)
The triangulation (4), K) of A  Rm gives rise to the triangulation (4), K') of A,
where K' is as in (1.8). We call (4),K') the barycentric subdivision of(4),K).
Here is the main triangulation lemma.
(2.8) LEMMA. Let (4),K) be a triangulation in R" of the closed and bounded de-
finable set A  Rm and let F be a full multivalued function on the triangulated set
(A,4>-1(K)). Then (4),K) or its barycentric subdivision (4),K') can be lifted to a
triangulation (IJ!, L) in R"+1 01 A F that is compatible with the sets in 4>-1 (Kt.
PROOF. Note first that K is closed. Replacing K if necessary by its barycentric
subdivision K' and F by the corresponding set of restrictions, we may assume in
addition to fullness of F that the following condition holds for each CE4>-1(K):
{ If 11, h E FIC, It 1 h, then there is at least one vertex of
C where clUd and cI(h) take different values.
(*)
Here the verticesofC are4>-1(au),.. .,4>-1(ak) with au,.. .,akthe vertices of4>(C).
Fix a linear order on Vert(K). We now construct Land ili above each C E 4>-1(K).
Let au,..., ak be the vertices of cI>(C) listed in the order we imposed on Vert(K).

130
TRIANGULATION
Let f < g be two successive members of FIG. Put
r;:= cI(J)(4>-l(ai»)' 8;:= cI(g) (4)-1 (aj»), b j := (a;,r;), Ci:= (ai,s;)ER'Hl
Because of (*) the conditions for lemma (1.10) are satisfied. Let L(I,g) be the'
complex in Rn+l constructed in that lemma, so I L(I, g); is the convex hu II of
{bo,...,bk,co,...,Ck}' Define the map 1J!1,g:[d(l),cI(gJ:-+IL(I,g)1 by
1J!1,g(x,tcl(l)(x)+(1-t)cI(g)(x») = t4>b(X) + (1-t)4>c(x), Ot 1,
where 4>b(X), 4>c(x) are the points of[b o ,.. ., bk; and ;co,.. ., Ck] with the same an: III'
coordinates with respect to b o ,..., b k and co,..., Ck as 4>(x) E [ao,..., ak] has with
respect to ao,.. ., ak' Clearly  I,g is a bijection, and it follows from ChaptPr Ii,
(1.13)(ii) that 1J! I,g is continuous.
We also define for each f E FIG the closed complex L(J) in R n +1: it consists of
the k-simplex (b D ,. ..,bk) with b j = (a;,cI(J)(cp-l(a;»)) as above, and all its fan's.
Then IJr J: r(cI(I») -+ IL(I)I is by definition the homeomorphism given by
WJ(x,cI(J)(x») = 4>b(X)
where cpb(X) is defined as before.
Now let L be the union of all complexes L(J, g) and L(I), for all G E cp-l (1\"). /I.
lengthy but routine argument using fullness of F shows L is a closed complex in
Rn+l, and that any two among the 1J! l,g'S and 1J! I's coincide on the intersection of
their domains. (In the interest of conciseness we leave here the numerous det.ails
to the reader.) Hence these maps glue to a continuous definable map W : At'
ILl. One verifies easily that 1J! is a bijection. Hence, by Chapter 6, (1.12). IjJ is
homeomorphism. Using the last remark of (1.11) it is routine to check that thp
triangulation (IJr,L) lifts (4),K). Finally, note that (IJr,L) is compatible with the
sets in 4> -l(K)F. 0
(2.9) TRIANGULATION THEOREM. Let S  Rm be a definable set, with definablr
subsets Sl, . . . , S k. Then S has a triangulation in R m that is compatible with tll,.,e
subsets.
PROOF. By induction on m, the case m = 0 being trivial. Suppose the triall-
gulation theorem holds for a certain value of m. To prove that each definahle
set S  Rm+l has a triangulation in R m +1 compatible with any given definahle
subsets Sl,' .., Sk, we first reduce to the case that S is closed and bounded. j,('t
jl:R-+( -1,1) be the definable homeomorphism t....... t/,f['+7J, and SiJo and S;' the
images of Sand S; under the homeomorphism (Xl,"" X m ) ....... (Jl(xI),..., Jl(X"';) :
Rm -> (-1, 1)m. It suffices to prove that the closure of SI', a closed and bounrkd
set in RmH, has a triangulation in R m +1 that is compatible with its subsets s,.
and Sf,. . ., Sf. So, adjusting our notations, we may as we)) assume from the start
that S is already closed and bounded. Under this assumption on S, let
T := bd(S) U bd(Sd U . .. U bd(Sk),

TRIANGULATION
131
sO dim(T) < m + 1 by Chapter 4, (1.10). Note also that T is closed and bounded.
By the good directions lemma (4.2) in Chapter 7 the set T has a good direction
vector. Choose a linear automorphism of Rm+1 that maps this good direction
vector to the vertical unit vector f".+1' To obtain a triangulation of 5 in Rm+1
compatible with 51,. . ., Sk we may replace S and the 5 j 's by their images under
this linear automorphism, which replaces T also by its image, so we may assume
(* )
the unit vector e m +1 is a good direction for the set T.
Under this assumption we shall prove that there is a triangulation of It'(S)  R m
in R m that can be lifted to a triangulation of 5 compatible with S1,..., Sk.
Set A := 11'(5) = It'(T), a closed and bounded subset of Rm. By (*) and cell
decomposition the set T is the disjoint union of r(J)'s for finitely many definable
continuous functions f on cells Aj that form a partition of A. By the inductive
hypothesis there is a triangulation (4),K) of A in Rm that is compatible with
each set Aj. The nonempty restrictions of the f's to the sets of 4>-l(K) form a
multivalued function F on (A,4>-l(K») such that reF) = T. Since T is closed,
lemma (2.6) implies that F is closed. Unfortunately, F may not be full.
Therefore we modify P as follows to F. Each f E F extends, first continuously to the
closure of its domain, and then, by the last remark of (2.3), further to a continuous
definable function j: A ---> R. Change T to t := union of the graphs r(i) for f E F,
and take a new triangulation (4), K) of A in R m that is compatible with all sets in
ij)-l(K), all setn (5n r<i)) and 1t'(5jnr(i») for f E P, i E {I,..., k}, and such that
the nonempty restrictions of the j's (for f E F) to the sets of 'P := 4>-l(K) form a
multivalued function F on (A, 'P). Clearly reF) = t. Since t is closed, l' is closed
by lemma (2.6), and therefore F is full. Hence by the last lemma we can lift (4), K)
or its barycentric subdivision to a triangulation (1Jr, Lo) of AP that is compatible
with the sets of 'PP. From the compatibility of (4), K) with the sets mentioned it
follows that if BE 'P and g E FIB then reg)  S, or reg) n 5 = 0, and similarly for
Si, i E {I,.. ., k}. Also, if BE 'P and g, h are successive members of FIB it follows
from Chapter 6, (3.3), that (g,h)  S or (g,h)n S = 0, and similarly for each Sj,
1  i  k. Hence 'PP partitions S and each Sj. Let L := {(1 E Lo: (1  w(S)}.
Then (1J!IS, L) is clearly a triangulation of S compatible with 51,.. ., 5k and it lifts
(4), K) or its barycentric subdivision. 0
(2.10) The following example shows decisively that the use of a linear automor-
phism (alias a linear change of coordinates) in the proof above cannot be omitted.
Something like the choice of a "good direction" seems essential for triangulation.
EXAMPLE. Let 5:= {(x,y,z)ER 3 : Y = xz, lxi, Izi  I}. Then, with 1t':R 3 -+R 2
the usual projection map we have 11'(5) = {(x,Y)ER 2 : lyl  Ix I  I}. We claim
No triangulation oflt'(5) can be Jifted to a triangulation of 5.

132
TRIANGULATION
To see this, consider a triangulation (op, K) of 71'(8), and suppose this triangulatiolJ
can be lifted to a triangulation (.p, K) of S. We shall derive a contradiction. Lp[
'P := op-l(K) and P := .p-l(K), so 'P = {7I'(A): A E P}. Let T := {(O,O, z) :
Izl  I}  8. Then T is a union of sets in P since (0,0) must be a vertex of P
(exercise). In particular, T contains a one-dimensional set A E P, corresponding to
a I-simplex under .p. Let a, bE T be the two vertices of A.
Since A is not open in 8, it is a proper face of a set B E P corresponding to it
2-simplex under ci> (exercise). Then dim(B - T) = 2, and 11' is injective on 8 - T.
so dim(7I'(B» = 2. Hence 71'(B) E 'P corresponds to a 2-simplex under OP, and t h(.
vertices of 71'(B) are the images under 11' of the vertices of B. But 71' collapses thp
distinct vertices a, b of B to the single point (0,0) in lI'(B), so lI'(B) cannot havp
three distinct vertices, contradiction.
(2.11) DIGRESSION 1: CRITERION FOR DEFINABLE ;QJ:IVALENCE.
Call definable sets A  Rm and B  Rn definably equivalent if there is a
definable bijection from A onto B. Clearly "definable equivalence" is an equivalclIlP
relation on the class of definable sets.
COROLLARY. Let A  Rm and B  R n be definable sets. Then
A and B are definably equivalent <=> dim(A) = dim(B) and E(A) = E(B).
PROOL The =?-direction follows from Chapter 4. Suppose now that A and /I
have the same dimension and the same Euler characteristic. By our triangulatioll
theorem we may also assume that A and B are finite disjoint unions of simplpx('s
in Rm and Rn respectively. For this case we prove a bit more:
CLAIM. A and Bare semilinearly equivalent, that is, there is a semilinear bijectiolJ
from A onto B.
Before starting the proof of this claim, first three general observations.
(1) A k-simplex is, for k > 0, the disjoint union of two k-simplexes ami it
(k - I)-simplex. (Exercise, induction on k.)
(2) Any two k-simplexes are semilinearly homeomorphic. (By remark in (1.2).)
(3) E(O') = (_I)k for a k-simplex 0'. (See exercise 1 below.)
PROOF OF CLAIM. By induction on k = dim(A) = dim(B). For k = 0 the s('(:; ..I
and B are finite of the same cardinality E(A) = E(B), and any bijection wil; do.
Let k > O. Write A and B as finite disjoint unions of simplexes:
A = 0'1 U . . . U 0' P U . . . U 0' q,
B = Tl U . . . U Tr U . . . U T.,

TRIANGULATION
133
where the O'i and Tj are of maximal dimension k for 1  i  P and 1  j  r, and
of lower dimension for i > p and j > r.
(sing observation (1) we may arrange that p = r, and that there is at least one
(k - I)-simplex among the O'i for i > p, as we)) as at least one (k - I)-simplex
among the Tj for j > p. By observation (2) there is a semilinear bijection between
0'1 U '" U O'p and TI U .. . U Tp. Also A' := O'p+1 U . . . U O' and B ' := Tp+1 U .., U T.
have both dimension k - 1, and the same Euler characteristic, by observation (3).
So by the inductive hypothesis, A' and B' are semilinearly equivalent. Hence A
and Bare semilinearly equivalent. 0
(2.12) DIGRESSION 2: NL'MBER OF DEFINABLE HOMEOMORPHISM TYPES.
Let us say that definable sets A S;; Rm and B S;; R n are definably homeomorphic
if there is a definable homeomorphism f : A -+ B. This defines an equivalence
relation on the collection of definable sets in Rm, m = 0,1,2, . . ., and a definable
homeomorphism type is simply an equivalence class for this equivalence relation.
An easy consequence of the triangulation theorem is that there arc only countably
many definable homeomorphism types. To see this, suppose that a (nonempty)
definable set A S;; R m is definably homeomorphic to IKI where K is a complex with
N vertices. Let el,..', eN be the basis vectors of RN.
CLAIM. A is definably homeomorphic to a union offaces ofthe simplex (el, . . ., eN)'
Just take a bijection V : Vert(K) --+ {el,. .., eN}, and let L be the complex with
Vert(L) = {el,.", eN} such that V: K -+ L is a vertex map whose inverse is also a
vertex map; then IVI: IK; -+ ILl is a definable homeomorphism, and the claim is an
obvious consequence.
(2.13) We close this section with some exercises. Exercise 3 is intended for readers
familiar with elementary model theory. Moreover, exercise 4 depends on the result
of exercise 3. Actually, in the next chapter we obtain essentially the same results as
stated in the exercises without any model theory. I include the exercise nevertheless,
since it illustrates a typical model-theoretic method.
(2.14) EXERCISES.
1. Let Am be the m-simplex in Rm spanned by 0, el, . . ., em, where el> . . ., em are
the standard basis vectors. Show that
Am+! = {(x,r)ER m +1: xEA m , 0 < r < 1- LXi}.
Derive from this that Am is an open cell in Rm, and that each m-simplex in R m is
definably homeomorphic to an open cell in R m .
2. Show that each bounded semilinear set is a polyhedron. More generally, let
51," ., Sk be semilinear subsets of a bounded semilinear set S S;; Rm. Show that

134
TRIANGl:LATION
there is a complex Kin Rm such that S = IKI, and each Si is a union of simplexC's
of J(. (The result of this exercise holds for any ordered field R.)
3. Let S  Rm+n be definable. Show that the sets Sa fall into only finitely
many definable homeomorphism types as the parameter a ranges over R m . MOIC'
precisely, show there is a definable map I: s..... R N (for some N) such that for earh
a E Rm the map la : Sa..... RN is a homeomorphism from Sa onto a union of facC's or
(el"'" eN)'
4. Let S S; Rm+n be definable. Show that there are a partition of Rm into definablp
sets AI,..., A.. and for each i = 1,..., k a definable set F;  Rn and a definablp
homeomorphism hi,: S n (Ai X Rn)'::' Ai x Fi such that the diagram
S n (Ai X R n )

h,
. Ai X F i
/
Ai
commutes. (Here the downward arrows indicate the obvious projection maps.)
Hint: use the result of exercise 3 in combination with Chapter 6, (2.5), exercise '2.
93. Definable retractions and definable continuous extensions
As one might expect, the triangulation theorem has numerous consequences, and
we develop a few here that are needed in later chapters.
(3.1) HOMOTOPIES, RETRACTIONS AND EXTENSIONs.
Let A  R m and B <:;; Rn be definable sets, and I, g: A..... B definable continuolIS
maps. A definable homotopy between I and g is a definable continuous map
H:A X [O,I;.....B such that I(x) = H(x,O) and g(x) = H(x,l) for all x in A. (I'
may help to view H as a "continuous" family of maps (H t )O:5,t9, with Ht : A ..... /J
given by Ht(x) = H(x,t), so I = Ho and g = HI,)
A definable set A is called definably contractible to the point a E A if there is
a definable homotopy H : A x [0, 1] -+ A between the identity map on A and the TTl a;>
A..... A taking the constant value a. (Such an H is called a definable contraction
from A to the point a.)
Note that for a point aER m the map H:Rm x [0, 1].....Rm given by
H(x,t) = (l-t)x+ta
is a definable contraction from R m to a. Since each cell is definably homeomorphic
to some R m it follows that each cell is definably contractible to each of its points.

TRIANGULATION
135
Let a definable subset A' of a definable set A  Rm be given, and denote by i A ,
the inclusion map A' -+ A.
A map r : A -+ A' is called a definable retraction if r is a definable continuous
map such that r( x) = x for all x E A'. So a definable retraction r: A -+ A I satisfies
roiA,=IA',
Next we define a definable contraction from A to A' to be a definable homotopy
II: A x [0,1] -+ A between l A and i A , 0 r, for a definable retraction r: A -+ A'.
This map r is then uniquely determined by r( x) = H (x, 1), and if we want to be
more explicit, we call H a definable contraction from A to r. If in addition
1I( a', t) = a' for all a' E A' and t E [0, 1] we call H a definable strong deformation
retraction from A to A'. Note that for a E A a definable contraction from A to
the point a is the same as a definable contraction from A to the subset {a}.
We also need one important construction in complexes:
(3.2) Let K be a complex.
Given a definable set A  IKI we define the star stK(A) of A in K to be the
union of all simplexes U E K such that cI(u) n A ::j:. 0. Clearly stK(A) is an open
neighborhood of A in IKi, and in fact the smallest neighborhood of A in IKI that
is a union of simplexes of K.
We can now formulate a basic result on the existence of nice retractions.
(3.3) PROPOSITION. Let K be a complex and La subcomplex of K, closed in K.
Then there is a definable retraction r : stK' (ILl) -+ ILl such that for each point
xEstK' (ILl) -ILl the open line interval (x, rex)) lies entirely in the simplex of K'
that contains the point x.
(3.4) REMARKS.
(1) We shall give an explicit definition of the retraction r.
(2) Given such r the map H:stK,(ILI) x [0, 1]-+stK,(lLI) given by
H (x, t) = (1 - t) . x + t . r( x )
is a definable strong deformation retraction from stK' (ILl) to ILl. The
image reS) of each simplex S  slK' (ILl) of K' is contained in cI(S).
(3.5) Before proving (3.3) we need some preparations. Let E := Veft(K) =
VerteR} For each vertex c E E we define a function A. : I K I -+ [0,1] by setting
fOf a point x E (eo,. .., ek), for a k-simplex (eo,..., ek) E K ,
A.(X) . - 0 if elt{eo,.. .,ek},
Ai if e = ei, whefe x = L Ajej, L Aj = 1,
with 0 < Aj for jE{O,. ..,k}.

136
TRIANGULATION
We call A. the affine coordinate function of K corresponding to the vertex
e. Note that A. is definable and continuous, and for each x E I K I we have
x == L A.(X) . e,
. E E
== L A.(X).
. E E
Now {b(O'): 0' E K } is the set of vertices of the first barycentric subdivision I'
of K, and so we also have for each 0' E K the ane coordinate function Ab(u) of
K corresponding to the vertex b(O'). For convenience we denote Ab(") by Au, so
A,,: IKI == I K I-- [0, 1] is a definable (continuous) function.
Next we introduce for 0' E K a "weight function" w,,: IKI-- [0,1] as follows:
(i) if U E L, then w" is identically 1,
(ii) if O'E K - L, then w" is identically 0,
(iii) if O'EL - L, then
. ( ) _ E"<TELAT(X)
W q X -  '
L..J"<T E K AT(X)
provided the denominator E"<T E K AT(X) is nonzero, and w,,( x) = 0 oth-
erwise.
(The summations are over the TEL with 0' < T, respectively over the T E K with
0' < T.) Clearly w" is definable, though it may not be continuous.
Now define /-I" : IKI-- [0,1] by /-I,,(x) := w,,(x)..\,,(x). Clearly Ji" is definable, and
by the following lemma Ji" is also continuous.
(3.6) LEMMA. Let 0' E L - L, and let S E K' be a simplex on which the functioTl
E"<T E K AT vanishes. Then A" also vanishes on S.
PROOF. Let S = (b(O'o),..., b(O'n)) with 0'0 < ... < Un a K-flag, O'n E K. SUpPOS('
..\" is nonzero at some point of S, hence at every point of S. Then 0' is aI!lOtl?;
0'0,' .., Un-I, Un, but U = O'n would imply 0' E K n L = L, contradiction. lIenn' (1
is among 0'0,.. ., O'n-l, in particular, n 2: 1. Thus 0' < O'n E K, so that
L AT(X) > 0,
"<T E K
for all xES. 0

TRIANGl:LATION
137
(3.7) LEMMA. For each xEstK' (ILl) there is UEL with JI,,,(x) > o.
PROOF. Let S E K' be contained in stK' (ILl). As above, S = (b(uo),..., b(u n ))
for some K-flag Uo < ... < Un, Un E K. Some face of S intersects ILl, hence I7k E L
for some k E {O,.. .,n}. Then we have for U = Uk: fl"(X) = >.,,(x) > 0 for all
xES. 0
We continue the proof of proposition (3.3). Suppose the complex K lies in Rm.
Then we define the (definable, continuous) map r:stK'(ILI)Rm by
(*)
rex) = L fl"(X)' b(u) ,'with U ranging over K .
Lfl"(X)
(3.8) LEMMA. reS)  cI(S) n ILl for each S E K' contained in stK.(ILI), and
rex) = x for xE ILl.
Thus r(stK' (ILl))  ILl and for x E stK' (ILl) -ILl the open line interval (x, r(x))
lies entirely in the simplex S E K' that contains the point x. So this lemma implies
that r is a retraction with the properties stated in proposition (3.3).
PROOF OF THE LEMMA. As in the previous lemma, let S = (b(uo),.. .,b(u n )) with
0'0 < '" < 17n, 17n E K. Take k E {O,. . ., n} maximal with Uk E L. Consider a
U E K such that JI,,, does not vanish identically on S. Then >." does not vanish
on S, so U = Ut for some t. If t > k the function w", and hence also fl" would
vanish identically on S. Thus t  k. This gives for each xES a convex combination
rex) = LO<i<k Cl'jb(Uj) with 0 < Cl'j E R for all i and L Cl'j = 1, as is easily checked.
Hence reS)  (b(O'o),.. .,b(Uk))  deS) n ILl. If S  ILl, then 17n E L, so k = n,
and by distinguishing the cases U E L, U E K - I, and 0' E I - L, we see that in each
case fl"IS = >'"IS. (For the case U E I - L, use the fact that K n I = L.) Thus
rex) = x for all xE ILl. 0
This finishes the proof of proposition (3.3). In combination with triangulability it
gives the following.
(3.9) COROLLARY. Let A be a definable closed subset of the definable set B 
Rm. Then there are a definable open subset U of B containing A, and a definable
retraction r:d(U) n BA.
PROOF. By (3.3) there are a definable open nei!!;hborhood V of A in B and a
definable retraction rv : V  A. Applying Chapter 6, (3.5) to the closed subsets
A and B - V of B we find definable U, open in B and containing A, such that
cl(U) n B  V. Now restrict rv to d(U) n B. 0

138
TRIANGULATION
Here is an attractive further consequence, generalizing (2.2). It will be needed <lR a
technical tool in the next chapter, on trivialization, to make a multivalued functioll
full.
(3.10) COROLLARY. Let A be a definable closed subset of the definable set B  H".
Then each definable continuous function f : A -+ R can be extended to a deft/Ill ,)!.
continuous function j : B -+ R. More generally, let f : A -+ C be a deft/Ill"!.
continuous map into a definable set C  Rn that is definably contractible to a po;/!!
c e C. Then f can be extended to a definable continuou. function j: B -+ C.
PROOF. Choose U and r as in (3.9). By Chapter 6, \3.8) there is a delinabll'
continuous function>. :B-+ [0,1] with>' -1(0) == A and >. - (1) == B - U. Finaliy. let
<I> be a definable contraction from C to c. Then we define the extension I: B -. C-
of f as follows:
I(x) .- <I>(J(r(x»), A(X)), for xecl(U) n B,
.- c,forxeB-U.
One verifies easily that I has the desired properties. 0
Notes and comments
Section 2 of this chapter follows close1y the proof of the semialgebraic trianguJa t io:\
theorem by Delfs and Knebusch in [11;. The main difference is that the !!;uod
directions lemma is only a partial substitute for the use of Koether normalizal ion
in their proof. The way out of this difficulty is to first establish lemma (2.2) a::d
use that to construct a full multivalued function as starting point for the inductivr
triangulation procedure. The idea of using multi valued functions and lemma CUi)
was suggested to me by reading some (unpublished?) lecture notes on subanalytir
sets by H. Sussmann (mid 1980s).
The claim in (2.11) that two bounded semilinear sets are semilinearly equivaiclit if
and only if they have the same dimension and the same Euler characteristic was
made to me in conversation by Schanue1j see [51] for related results.
Section 3 of this chapter is adapted from the semialgebraic case treated by Delfs <I lid
Knebusch in [13], which contains further interesting material that extends eaily f()
our setting.
See Dieudonne [16] for some history of triangulation results for manifolds and <11-
gebraic varieties, and Giesecke [27], Lojasiewicz [39] and Hironaka [31] for variolls
kinds of triangulation theorems for semialgebraic and semianalytic sets.
The inductive way triangulation was established in this chapter provides extra ill-
formation that has been used by Woerheide in [65] to prove the excision property Ic))"
simplicial and singular homology for definable sets in the o-minimal setting. (Tltis

TRlANGULATIO:-I
139
way of obtaining the excision property seems to be new also in the semialgebraic
case. This case is extensively discussed by Knebusch in [34].)

CHAPTER 9
TRIVIALIZATION
Introduction
A continuous map f : S -+ A between topological spaces is often thought of as
describing a "continuous" family of sets U- 1 (a))aeA parametrized by the space A.
From this viewpoint the trivial maps are the simplest. One says that f is trivial if
f looks like a projection map A x F -+ A, precisely, if there are a topological space
F and a homeomorphism h: S -+ A x F such that the following diagram commutes.
S - . AxP
h/
A
Kote that then all fibers f-l(a) are homeomorphic to P, and that h = (f,).,) for
some unique continuous map).,: S -+ F. We also say that f is trivial over a subspace
B of A if the restriction flf- 1 (B):f- 1 (B)-+B is trivial. (Then all fibers f-l(b)
with bE B are mutually homeomorphic.)
As in the previous chapter we fix an o-minimal expansion n = (R, <, S) of an
ordered (necessarily real closed) field (R,<,O,I,+,-,.).
The main result of the first section of this chapter generalizes a well-known theorem
of Hardt [29] on semialgebraic continuous maps between real semialgebraic sets, and
is roughly that if f, S, and A as above are definable-in the sense of n of course-
then one can partition A into finitely many definable subsets over each of which
f is trivial. This is even true in the stronger sense that the correspondinl!; F's
and h's can also be taken to be definable. This result was already established in
Chapter 8, (2.14), exercise 4, but there it depended on a model-theoretic argument.
The constructive proof given here is more elementary, though much longer. We
also givp some further refinements, which in Section 2 lead to applications like the
conical local structure of definable sets. In Section 3 we show how Wilkie's theorem
used in combination with our trivia.lization results implies new topological finiteness
properties of polynomials with few terms. We solve in particular a problem of Risler
and Benedetti on semialgebraic sets of bounded additive complexity. In this way
we continue a theme initiated by Khovanskii [32,33].
141

142
TRIVIALIZATION
1. Trivialization theorem
(1.1) Consider a definable map 1: S -+ A, where A  R m and S  R n are definabl"
sets. Viewing A as a base space or parameter space, f describes the family 0:'
sets (1-1 (a») a€A' A definable trivialization of 1 is a pair (F, A) consistinf!; 0'
a definable set F  RN, for some N, and a definable map A : S -+ P such that
(1, >.) : S -+ A x F is a homeomorphism.
So (1, >.) identifies S with the cartesian product A X F, and under this identificatjolJ
1 corresponds to the projection map A x F -+ A. Note that then 1 and >. arp
continuous, and that (1, A) maps each fiber 1-1(a) of 1 homeomorphically onto
{a} x F, in particular all fibers are definably homeomorphic to F.
We call 1 definably trivial if 1 has a definable trivialization. Given a d"
finable subset A'  A we call 1 definably trivial over A' if the restrictiolJ
1If- 1 (A'): 1- 1 (A') -+ A' is definably trivial. Note that if I is definably trivial.
then I is definably trivial over each definable subset of the base space A: if (F, .\)
is a definable trivialization of f and A'  A is definable, then (F, AI/- 1 (A')) is a
definable trivialization of 11/- 1 (A'):/- 1 (A')-+A'.
We can now formulate the main theorem of this section.
(1.2) THEOREM. Let I: S -+ A be a continuous definable map as above. Then thfTf'
is a finite partition A = Ai U . . . U AM of the base space A into definable sets Ai
such that f is definably trivial over each Ai.
We actually provide a stronger version where the definable trivializations over "af'll
Ai are also compatible with distinguished definable subsets of S.
First we establish an easy lemma on good directions. Next we prove a technical
lemma involving homotopies and retractions, and then we can trivialize.
(1.3) Let n > 0, and sn-1 := {x ERn : IIxll = I}, the set of unit vectors. Recall
that a unit vector u E sn-l is a good direction for a set T  R n if each line wit h
direction u intersects l' in only finitely many points. (A line with direction '/1 is a
set p + R'/1 with pE R n .) Otherwise u is a bad direction for T.
If T is definable and dim(T) < n then it folluws from the good directions lemma
(4.2) in Chapter 7 that almost all unit vectors u E sn-l are good directions for 1.
in the sense that the set of bad directions is of dimension < n - 1.
(1.4) LEMMA. Let T  R m + n be definable, dim(T) < m + n, n > O. Then th(/,f
exists a unit vector u E sn-l such that the set
{a E R m : u is a bad direction for Ta}
has dimension < m, that is, u is a good direction lor almost all sets Ta.

TRIVIALIZATlO:-I
143
PROOF. Note that dim {a E R m : dim(T.)::;:: n} < m, so we may as well discafd
from T the subsets {a} x T. for which dim(T.) ::;:: n. Hence we have reduced to the
case that dim(T.) < n for all a E R m . FOf each a E Rm let B.  sn-1 be the set of
bad directions for T.. Then dim(B.) < n -1 fOf all aE R m , so dim(B) < m + n-l
where B:::;:: U{{a} X B.: aER m }  Rm X sn-l. Let p:Rm x sn-l sn-1 be
the projection map onto the second factor. Then it follows from Chaptef 4, Section
1 that thefe is u E sn-1 such that dim (p-l (u) n B) < m. This vectof u is a good
direction for almost all To. 0
The following technical lemma is the key to the trivialization theorem. To formulate
it we need a mild generalization of cells: define a generalized open cell in R m to
be a definable opeD subset of Rm that is definably homeomotphic to R m . Examples
are open cells in R m and m-simplexes in Rm. Clearly each generalized open cell is
definably contractible to each of its points.
(1.5) LEMMA. Let 8 1 "", Sk  Rm+n be definable, let A <:;; Rm be definable, and
let 7!" : Rm+n  R m be the projection map. Then there are disjoint generalized
open cells Ai, . . ., Ah <:;; A such that dim (A - U Ai) < m, and such that for each
i E {I, . . . , h} and each definable contraction H : Ai X [0, 1]  Ai from Ai to a point
a E Ai there exist a definable retraction r : Ai X Rn  {a} X Rn and a definable
contraction iI : Ai X Rn X [0, 1]  Ai X R n from A, X Rn to r, with the following
properties:
(1) illifts H, that is, the following diagram commutes:
Ai X Rn X [0,1] If Ai X R n
-
",Xid 1 1'"
Ai X [0,1] ---> Ai
H
where 7!"i ::;:: 7!"IAi X R n .
(2) For each Sj the maps rand iI yield by restriction a definable retraction
rj:(Ai X R n ) n Sj- ({a} X R n ) n 5j
and a definable contraction
ilj:(A i X Rn)nSj) X [O,I]-(A i X Rn)nSj
from (Ai X Rn) n Sj to rj.
(3) For each x E A, the map r restricts to a definable homeomorphism
r:z::{x} X Rn{a} X R n
that maps ({x} X Rn) n 5 j onto ({a} X Rn) n Sj for each j.

144
TRIVIALIZATION
PROOF. By induction on n. The case n = 0 follows by cell decomposition. In llJOfP
detail: take a finite partition of A into cells that is compatible with SI,.'" Sk.
and let AI,. . ., A" be the ope!! cells in this partition; then H and the point a l Ai
determine r, and by taking H = H the conditions (1), (2), and (3) are trivially
satisfied.
Let n > 0 and assume inductively that the assertion holds for n - 1 instead of n.
We then first do the case that 51"." Sk are all bounded. Let
T := U bd(Sj),
so that T is closed and bounded, and dim(T) < m + n. (Later we modify T to
a set t that is the graph of a full multivalued function.) By the previous lemma
there is a unit vector u E sn-l that is a good direction for almost all sets Ta, t II at
is, dim(B) < m where B:= {aER m : u is a bad direction for Ta}.
Recall that el,..., en denotes the standard basis of the vector space Rn over ii.
We view en as pointing in the "vertical direction". Take a linear automorphism
)., : Rn -+ Rn such that ).,( u) = en, and replace each set S j by its image {(x, A( Y)) :
(x,Y)ESj} under the automorphism idx >. of Rm+n. This is permitted sinn- a
solution to our problem for the new Sj's can be transformed back via id x A-I tu
a solution of the problem for the original S/s. Note also that then T gets replacpd
by {(x, >.(y)): (x, y) E T}, so our new T satisfies
dim(B) < m, where B = {aER m : en is a bad direction for Ta}.
Now let p: Rm+n -> R m + n - l be the projection map on the first m+n-l coordinatps
and q : Rm+n-l -> R m the projection map on the first m coordinates, so that
11' = q 0 p. Then e m + n is a good direction for T n 11'-1 (A - B). Hence by n,1I
decomposition the set Tn 'II'-I(A - B) is the disjoint union of r(J)'s for finitE'ly
many definable continuous functions I defined on cells C that form a finite partition
P of (A - B) x R n -l. Call these functions on cells C E P distinguished. Take
a triangulation (,K) of (A - B) x Rn-l compatible with each cell C E P. L!'t
F be the collection of the restrictions liD of the distinguished functions I on
cells C E P to sets D E -I(K) with D  C. Then F is a rnultivalued function on
((A - B) X Rn-l, -I(K)) in the sense of (2.5) of Chapter 8, and F is closed by (Vi)
of Chapter 8, since reF) = Tnll'-1 (A - B) is closed in 1I'-I(A - B) = (A- B) X 17".
Unfortunately, F may not be full, and therefore we modify F as follows to a full
multivalued function P. First extend each I E FID continuously to the closurE'
cI(D) n (A - B) of D in A - B, and then further to a bounded continuous definabk
function i:A - B-+R, usinl!; (3.10) of Chapter 8. Now put
t := ( U rei) ) u ruo),
JeF
where 10: (A - B) x R n - l -> R is the function that is identkally O.

TRIVIALIZATION
14!;
Next replace (,K) by a new triangulation (,k) of (A - B) x RTI-l that is
compatible with all sets in -I(K), all sets p(Sj n r(i)) for IE F, and all sets
p(Sj n ruo)). In addition we arrange that the nonempty restrictions of the j's
for 1 E F and of 10 to the sets in -l(k) form a multivalued function F on the
triangulated set ((A - B) X RTI-l, -l(lr)).
Note that f(F) = t, that t is closed in (A- B) X Rn, and that Tn ((A- B) x Rn) 
t. Clearly F is full. Let FID = {JDl,...,1Dk(D)} with IDl < ... < 1Dk(D),
k(D)  1, where D E -l(k). The compatibility of -l(k) with the sets indicated,
together with (3.3) of Chapter 6, implies that for any 1, 9 E FID and set Sj we have:
either f(J)  Sj> or f(J) n Sj = 0;
if 1 < 9 are successive members of FID,
(*) then either (1, g)  Sj, or (1, g) n Sj = 0;
if 1 = 1Dl is the first member of FID, then (-00,1) n Sj = 0;
if 9 = 1Dk(D) is the last member of FID, then (g, +00) n Sj = 0.
f'ow apply the inductive hypothesis to the sets in -l(k) and A - B instead of A.
This gives disjoint generalized open cells AI, . . ., A"  A - B with dim (A - U Ai) <
m and with the following properties for any given i E {I, . . ., h}:
(1)' Each definable contraction H: Ai X [0,1] -> Ai from Ai to a point aE Ai can
be lifted to a definable contraction H': Ai X R n - 1 X [0,1] -+ Ai X Rn-l from
Ai X Rn-l to a definable retraction r': Ai X R n - 1 ..... {a} x Rn-l.
(2)' For each D E -l(k) the maps r' and H' from (I)' yield by restriction
a definable retraction rb : (Ai X Rn-l) n D -+ ({a} X Rn-l) n D and a
definable contraction Hb : [(Ai X R"-l) n D] X [0,1] ..... (Ai X Rn-l) n D
from (Ai x R"-l) n D to rb.
(3)' For each x E Ai the map r' from (1)' restricts to a definable homeomorphism
r: {x} xRn-l.....{a} XR,,-l mapping ({x} X Rn-l)nD onto ({a} X Rn-l) n
D, for each DE-l(k).
Fix an Ai and write Ai X Rn as the disjoint union of the following sets:
GDk'- {(Y'/Dk(Y)): YE(AiXRn-1)nD}, DE-l(k), l$k$k(D)j
UDk {(V,s): YE(AixR"-l)nD, 1Dk(y)<s<IDk+l(Y)}'
DE-l(k), 1 $ k < k(D):
U D - .- {(y,s): YE(Ai X Rn-l) nD, s < 1Dl(Y)}, DE-l(.k);
UD+ .- {(y,s): YE(Ai X Rn-l) nD, s> IDk(D)(Y)}, DE-l(k).
By (*) every set (Ai X R") n Sj is a union of some of these sets.
Let a definable contraction H: Ai X [0,1] -> Ai from Ai to a point a E Ai be given,
and let H' and r' be as in (1)', (2)' and (3)' above. We now show how to lift H' to

146
TRIVIALIZATION
a definable homotopy il: Ai X R n x [0, 1] -> Ai X Rn by defining il on each of the
sets Gm x [0,1], Um x [0,1], UD- X [0,1] and UD+ x [0,1]:
(i) il(y,im(y),t):= (H'(y,t), !Dk(H'(y,t))),for (y,!Dk(y))EGDk, tErn, I);
(ii) write each z E U Dk uniquely as z = (y, (1 - U)!Dk(Y) + U!Dk+1 (y)) wil h
0< U < 1 and set, for tE[O, 1],
il(z, t) := (H'(y, t), (1 - u)!m (H'(y, t)) + U!Dk+dH'(y, t)));
(iii) write each z E UD- uniqueJy as z = (y,JD1(Y) - u) with u > 0 and S('1
N(z, t) := (H'(y, t),fm (H'(y, t)) - u), for t E [0, 1];
(iv) write each zE UD+ uniquely as z = (Y,!m(D)(Y) + u) with u> 0 and s"1
N(z, t) := (H'(y, t), fDk(D) (H'(y, t)) + u), for t E [0,1].
Clearly N maps GDk x [0, 1] into GDk, UDk x [0, 1] into UDk, UD- X [0, 1] into u[)_.
and U D+ X [0, Ij into U D+. A lengthy but routine argument using fullness of l' (and
for instance lemma (4.2) from Chapter 6) shows that il is continuous. HencE' Ii is
a definable contraction from Ai X Rn to the retraction r: Ai X Rn -+ {a} x Rn gi vl'n
by r(z) := il(z, 1). One checks easily that il and r have the properties requirl'd
in (1) and (2). To check (3) one shows first that for fixed y E Ai X Rn-l we hit\.1'
r(y,s) = (r'(y),gy(s)), where gy: R -+ R is a strictly increasing bijection. Tlllh
each map r:z:: {x} x Rn -+ {a} x R n as in (3) is bijective. To get the continuity of
the inverse of r", one first shows there is a constant M > 0 in R, independent of
y E Ai X Rn-1, such that Igy( s) - sl  M for all s, and then one applies ICUlllla
(4.2) from Chapter 6, and the inductive assumption (3)' on r.
This settles the case that Sl, . . . ,Skare bounded. !\ ow we reduce the general ca,,'
to this special case. Let [ be the interval ( -1,1) and Ji: R -+ 1 the horneornorplti""
given by p.( x) := x / Y'f+'Xf, and let flp: RP -+ JP be the homeomorphism give" h'
Jip(Xl,.. .,xp):= (Ji(xd,.. .,p.(x p )). Note that Jip is definable.
Let definable Sl,..., Sk <::;; Rm+n and definable A <::;; R m be given. Then the ;,('1 s
SI'; := /lm+n(S;) <::;; R m + n are bounded, and we add to Sl'l,' .., Sl'k one extra ,,'1.
namely SI'O := [m+n, so SI'O = p.m+n(Rm+n). Put A" := Jim(A). We now app!y
the result proved for the bounded case to the sets S ,,0, . . ., S "k, with A" play: "1-\
the role of A. This !!;ives us disjoint generalized open cells Al'l" . ., A"h <::;; A" wit;I
dim(A" - U A"i) < m such that for any given i E {I,.. ., h} we have:
(1,,) each definable contraction H,,: A,,; x [0, 1] -+ A,,; to a point a" E Al'i lift s 1 ()
a definable contraction if" : Al'i X Rn x [0, 1] -+ A "i X R n from A"i x H"- t ()
a definable retraction rl': Al'i X R n -+ {a,,} x R n ;
(2,,) for each S"j (including S"o) the maps r" and il" from (11') yield by rcstr;,-
tion a definable retraction Tl'j: (Al'i X Rn) n Sl'j -+ ({a,,} X Rn) n S"j, and
a definable contraction ii "j: ((Al'i X R n ) n Sl'j) x [0,1] -+ (A". X R n ) n S,,)
from (A,,; x Rn)n S"j to Tl'j;
(3,,) for each x E A"i the map r" from (11') restricts to a homeomorphism Tl'l:
{x} x Rn-+{a,,} x Rn that maps ({x} X R n ) n Sl'j onto ({al'} X R n ) n 8,,)
for each j.

TRIVIALIZATION
147
Xow we use Ji;.1 : 1 m  R m and Ji;;"n : fm+n  R m + n to transform back. Put
A.:== J1;.I(Al'i), i == 1,...,h. Clearly Al,...,A" are generalized open cells and
dim (A - U Ai) < m. Let II: Ai X [0,1]  {a} be a definable contraction from Ai to
a point a E Ai. We shalllind T and il with properties (1), (2), and (3) of the lemma.
Set al' :== Jim(a) and define lIl': AI" X [0,1] ---> Al'i by H I'(x, t) :== J.tmH (p.;.1 (x), t).
Then HI' is a definable contraction from Al'i to aIL' Let HI' and TI' be as in
(11')' (21')' and (31') above. Applying (21') to SI'O == Im+n we see that HI' maps
Al'i X 1 n X [0,1] into Al'i X In, so we may define jj :Ai X Rn X [0, 1] Ai X Rn by
H(x,y,t) := p.;;"nill'(J1m(x),J.tn(y),t).
Similarly r I' maps AI" X 1 n into {a I'} X In, by (2) applied to S 1'0, so we may define
r :A i x R n -+ {a} X R n by r(x,y) := J1;;'"rl'(J1m(x)'J1n(Y»). One verifies easily
that then il and r satisfy the statements of the lemma. 0
(1.6) In our trivialization theorem we also want to take into account distinguished
definable subsets of S. Here are the relevant definitions.
Let A  Rm and S  R" be definable sets, f: S - A a continuous definable map,
and SI, . . ., S k definable subsets of S.
Then a definable trivialization of f respecting SI, . . ., Sk consists of a definable
trivialization (F, >.) of f and distinguished definable subsets Fl,.. ., Fk of F, such
that (f, >')(5.) = AxF i for i = 1,..., k. Note that then each restriction jlSi: S; ---> A
has definable trivialization (Fi, >'IS;). Given definable A'  A we call (fj St,.. ., Sk)
definably trivial over A' if the restriction
flr l (A ' ) :r l (A / )---> A'
has a definable trivialization respecting SI n f-l(A'),.. ., Sk n 1-1 (A'). :'I'ote that
then (lj SI,"', Sk) is also definably trivial over each definable subset of A'. Also,
if a E A' we can always take a definable trivialization over A' respecting SI n
1-1 (A'),. .., 5k n 1-1 (A') to be of the form (J-l(a), >.) with>' the identity on
/-1 (a)  1-1 (A'), and with distinguished subsets SI n I-l( a),. . ., Sk n j-l (a) of
j-l(a).
(1.7) TRIVIALlZATIO:i THEOREM. With I:S-A and SI,', .,Sk as above, we can
partition A into definable subsets AI, . . . , AM, such that (J; SI, . . . , S k) is definably
trivial over each A..
PROOF. By induction on dim(A). If dim(A) :5: 0, then A is finite, and the theorem
holds trivially. Let dim(A) > 0, and assume the desired result holds for lower
values of dim( A). By partitioning A into finitely many cells A', and working with
the restrictions 11/- 1 (A') : 1-1 (A') ---> A' we may as well assume that A is a cell;
then using the homeomorphism PA from A onto an open cell, we may further
assume A is an open cell, say in Rm, so dim(A) = m. f'ext we replace S by the

148
TRIVIALIZATION
reversed graph of 1 by applying the definable homeomorphism V...... (f(V), V) frOtn
5 onto this reversed graph. So from now on 5 is a definable subset of R"'+>'
and 1 is the restriction 1/'15: 8 ..... A of the projection map 11' : Rm+" ..... Rm. w(:
need one further reduction to the bounded case. Let Ji ; R..... I = (-1,1) be the.
definable homeomorphism given by Ji( x) = x / JIt:X2, and let /-Ip; RP ..... IP be I.h0
induced homeomorphism given by Jip(Xl'."'X p ) = (Ji(xt)"",/-I(x p »), for P(]\i.
Put 5p. ;= Jim+n(5), 8/J.i ;= /-Im+,,(8j), Ap. := Jim(A), and let Ip.; 5p...... A/J. be thp
restriction of the projection map R m +"..... R m . Clearly it suffices to show
(*) { Ap. can be partitined into defin.ale subsets Ap.l,. . ., A/J.M such that
(lp.; 8p.1,. .., Sp.k) IS definably trlVlal over each Ap.i.
We distinguish one more subset of R m +", namely Ap. x d(I)", where I = (1, I).
and apply lemma (1.5), with 8/J.,5p.1,...,8p.k and Ap. x cl(l)" as distingllish"d
subsets of R m +", and Ap. in the role of A: this gives disjoint generalized open ('(':is
A/J.I"'" Ap.h in Ap. with dim(Ap. - U AP.i) < m, and further properties from leml/la
(1.5). By the inductive assumption we can partition Ap. - U Ap.; into finitely many
definable subsets over each of which (lp.; 5p.1" .., Sp.k) is definably trivial. So (+)
will follow from
(**) (lp.i8p.1," .,Sp.k) is trivial over each Ap.;, 1 ::; i::; h.
Pix an i and take a definable contraction H from Ap.; to one of its points a. Thpn
lemma (1.5) gives us a definable retraction r;A/J.i X R".....{a} X R", which for ea('h
x E A /J.i maps {x} x R" homeomorphic ally onto {a} X R", {x} X S p.:r: onto {a} x S"."
{x} X 8p.j:r: onto {a} >< Sp.jo (1 $ j ::; k), and {x} X cl(!)" onto {a} x d(!)".
Write rex, V) = (a, sex, y», where s ; Ap.i X Rn ..... R" is a definable continuous
map. Then the map (x, V) ...... (x,s(x,V») : Ap.; x cI(I)n ..... Ap.; x cI(I)" is ,\
continuous definable bijection. This bijection must be a homeomorphism: cef'll
point in Ap.; has a definable neighborhood N in A/J.i that is closed and bounded, aud
this bijection maps N x d(I)n homeomorphic ally onto itself, by Chapter 6, (1.12;
Define Ai; 1;:1 (A/J.;) ..... 5p.a by Ai(X,V) ;= s(x,y). It is now easy to verify that.
(Sp.o, Ai) is a definable trivialization of (I/J.; 8/J.!,'''' Sp.k) over Ap.;. This finishps
the proof of (**). 0
(1.8) Note that theorem (1.2) is the special case k = 0 of theorem (1.7). Readprs
familiar with elementary model theory will note that the easy model-theoretic ar-
guments (in Chapter 8, (2.14» proving theorem (1.2) can be used to derive theor('1II
(1.7) in a similar way.
(1.9) The following exercise gives an alternative formulation of the trivialization
theorem, and explains why such results are also referred to as "generic" triviality
theorems.
EXERCISE. Let I; 5..... A be a continuous definable map between non empty defill-
able sets 5 and A. Show that there is a definable set E  A with dim E < dim A
such that 1 is definably trivial over each definably connected component of A - L';.

TRIVIALIZATION
149
2. Applications
(2.1) An immediate application of the trivialization theorem is that for a given
definable set S  Rm+n the definable sets S., fall into only finitely many different
definable homeomorphism types as x ranges over R m .
To see this, let f: S -+ R m be the restriction of the projection map Rm+n -+ R m ,
and take a partition of Rm into definable sets AI,..., AM over each of which f
is definably trivial, say with definable trivialization (Ai, F i ) over Ai. Then Sx is
definably homeomorphic to F i for all x E Ai'
A (well-known) special case ofthis finiteness result is the fact that for fixed natural
numbers d and n the zero sets of polynomials f(XI,...,X n )EJR[Xl,''',X n ] in
an of degree at most d fall into only finitely many semialgebraic homeomorphism
types. (Instead of JR we can take here of course any real closed field.)
We can strengthen these results by considering "embedded homeomorphism type" .
Two sets X, Y  JRn have the same embedded homeomorphism type if there
is a homeomorphism h : IItn ..::+ IItn such that h(X) = Y. To have the same
embedded homeomorphism type is an equivalence relation on the collection of sub-
sets of JRn. Similarly we say that two definable sets X, Y  R n have the same
embedded definable homeomorphism type if there is a definable homeomor-
phism h : R n ..::+ R n such that h(X) = Y. To have the same embedded definable
homeomorphism type is an equivalence relation on the collection of definable sub-
sets of Rn. Using the "triviaJjzation theorem with distinguished subsets" the same
arguments as above show
PROPOSITION. If S <:;; Rm+n is definable, then the definable sets Sx  Rn fall into
only finitely many embedded definable homeomorphism types, as x ranges over R m .
(2.2) Next we give an application to the local structure of definable sets. Given a
definable set A  R n and a point p E R n not in A we define the cone with base A
and vertex p to be the set [A, p] :== {ta + (1- t)p: a E A, 0:::; t :::; 1}, the union
of the line segments [a, p] with a EA.
Given pERn and f > 0, we also put
B(p, f) .- {x E R n : Ilx - pi I :5 f}, the closed ball centered at p with radius f;
S(p, f) ._ {x ERn: IIx - pi I = f}, the sphere centered at p with radius c.
(2.3) THEOREM. Let E <:;; R n be a definable set and p a non-isolated point of E.
Then there is £ > 0 such that En B(p, f) is definably homeomorphic to the cone with
base En S(p, £) and vertex p. More precisely, there is a definable homeomorphism
rj> from B(p, €) onto itself, such that
(i) rj>(p) = p and rj> is the identity on S(p,€),

150
TRIVIALIZATION
(ii) !lrt>(x) - pll = Ilx - pll for all xEB(p,£),
(iii) rt>(E n B(p, f») = [E n S(p, £),p].
PR.OOF. Apply the trivialization theorem to the distance function x ,..... IIx - pil :
Rn..... [0, 00), with E as a distinguished subset of R n . Note that for £ > 0 the (-Ii hpf
of this map is S(p, f), and that the inverse image of (0, f] is B(p, f) - {p}. This giv<"
a definable trivialization >':B(p,f)- {p}.....S(p, f) respecting E over a half-intl'eval
(0, (], ( > 0, with >'IS(p, () = identity.
Hence we have a definable homeomorphism
x..... (lix - pl;,>.(x»):B(p,f) - {p}.....(O,f] X S(p,£)
mapping En (B(p, f) -{p}) onto (0, f] X (E n S(p, f»).
Now define 1/1: B(p, f)..... B(p, f) by rt>(x) = p + (lix - pll/f) . (>.(x) - p) for x f- ]1.
and ljJ(p) = p. One verifies easily that this <p has all the desired properties. lJ
3. On a conjecture of Benedetti and Risler
(3.1) The generic triviality theorem leads to a new result on zero sets of "!'ewllo-
mials", as recorded in the following proposition. We will make essential use of IiI('
o-minirnality of the model-theoretic structure R.xp:= (R,<,O,I,+,-,.,exp) dlle
to Wilkie [64]; see also [22] for another proof that Roxp is o-minimal.
(3.2) PROPOSITION. For any given natural numbers m and n there are only finite Iy
many homeomorphism types among the sets Z (1) := {x ERn: f( x) = O}, 11'111 rc
I(X l ,..., X n ) E R[X!,. .., Xn] has at most m monomials.
PROOF. We are of course going to use the fact that the function
(x,y)..... x Y := exp(ylog(x») :(0,00) X R.....R
is definable in Roxp. A minor complication is that, on the other hand, there is 110
function F :R2..... R definable in Roxp such that F(x, k) = xk for all x E Rand k ( N.
(See exercise (3.9).) Fortunately we can instead define two functions E,O :R2..... R
in Roxp such that E(x,k) = xk for all xeR and ke2N and O(x,k) = xk for "II
x E Rand k E 2N + 1. We define E and 0 as follows:
{ Ix:Y for x #- 0,
E(x,y):= 1 forx=y=O,
o for x = 0, y #- OJ
{ xY for x > 0,
O(x,y):= -(-x)Yforx<O,
o for x = O.

TRIVIALIZATJON
151
Let us now fix natural numbers m and n, and let f = ;';,1 aiX" denote an arbi-
trary polynomial in X = (Xl,"" X n ) over JR with at most m monomials: ai E JR
and ai = (ail> . . ., ain) E Nn for i = 1,. . ., m. Let
Even;(J) := U: 1 5: j 5: n, ai) E 2N} and Odd;(f) := {I, .. ., n} - Eveni(J)
for i = 1, . . ., m. Then we have for all x E JR n
f(x) = Lai IT E(Xj,aij) IT O(Xj,aij).
j E Even,(J) j E Odd.(1)
In this expression we may consider the (m + mn)-tuple
c(f):= (a1," .,am,Ctll,.. .,a1"," "ml>" .,a mn ) EJRm+mn
as a parameter. Let JEt,..., EK, with K = 2 mn , be the different n-tuples of
the form (Even1(f),..., Evenm(f)} as f varies. The considerations above show
that there are sets Sl'''', SK  JRm+mn X JRn, definable in JR exp , such that if
(Even1(J),...,Evenm(J») = Ek, 1 5: k 5: K, then Z(f) = (Sk)c(I)' (In other
words, the zero sets of the real polynomials in X = (Xl,"" X n ) with at most m
monomials fall into 2 mn different definable families, where "definable" here means
"definable in JR exp ".) The desired result now follows from the fact that among the
subsets of JRn of the form (Sk)c (1 :$ k :$ K, c E R m+mn) thefe afe only finitely
many homeomorphism types, by (2.1). 0
(3.3) REMARK. The same afgument shows that for given m, n there afe only
finitely many different embedded definable homeomorphism types among the sets
Z(f)  R n of pfoposition (3.2), where "definable" refers to definability with pa-
raIneters in Rexp. (Here we use the pfoposition stated at the end of (2.1).)
(3.4) From a geometric vjewpoint the "number of monomials" is not a good com-
plexity measure for polynomials, fOf example, it misbehaves under linear changes
of variables. Therefore Benedetti and Risler [2] introduced the more geometric no-
tion of "additive complexity", which also makes sense for rational functions. Let
us say that a rational function f E JR(X), with X = (Xl"'" X n ), has additive
complexity :$ kif thefe are nonzero rational functions It, . . .,!k E JR(X) such that
It = a1X u , + b 1 X v i
with aI, b l E JR, U1 = (un,. . ., Ul n ) E Z n, V1 = (Vll, . . ., V1n) E zn, and for 1 < j :$ k
j-1 j-1
h = ajXUj IT f?' + bjXVj IT f? ,
i=l i=l
with aj, bjEJR, Uj = (Ujl>"', Ujn) EZ", Vj = (Vjl>"" Vjn) EZ n , rji EZ, 8ji EZ fOf
1 :$ i < j, and
k
f = cX Ui IT f?
j=l

]52
TRIVIALIZATION
with cEJR, w = (w1,...,w n )EZ n and tl,...,tkEZ. In other words, starting \\ilh
the variables Xl,"" X n and real constants and allowing addition, multiplication
and division one can obtain f using at most k additions (and an unlimited number
of multiplications and divisions.)
Let us say that a semi algebraic set A  R n has additive complexity at mOst
(n,p,k)EN 3 if
A = U Ai, where I is a finite index set and for each i E I
iEI
Ai = {xEJR n : J;(x) =... = f;;(x) = 0, 9(X) > O,...,g,(x) > O},
with polynomials f, 9  E JR[ X], all of additive complexity at most k,
and L(qj + r;}  p.
iEI
Then we have the following finiteness result conjectured by Benedetti and Jtj,ll'r,
see pp. 214-215 of [2].
(*)
(3.5) THEOREM. Given any triple (n, p, k) E N3 there are only finitely many riif
ferent embedded homeomorphism types of semialgebraic subsets of Rn of addllie,
complexity at most (n, p, k).
Note that this generalizes proposition (3.2) above on zero sets of fewnomiak As
with that proposition, the theorem we just stated follows from the fact th a!,
given (n,p,k), there are sets Sl,,,.,SM  RN+n (for suitable M,N depcndiIl!!;
on (n,p, k), each definable in Rexp, such that each semialgebraic set A  R" 0; ad-
ditive complexity at most (n,p, k) is ofthe form A = (Sm)c for some mE {I,. ... ,H}
and some c E JRN. To prove this fact we may assume that not only (n, p, k) is gin'n,
but also the finite index set I and the natural numbers qi (i E I) and ri (i E /)
appearing in the representation (*) of the semialgebraic sets A we are considf'fing.
The desired result will then follow easily from the next two lemmas by the same
kind of argument as used in the proof of (3.2).
(3.6) LEMMA. A rational function f E JR(X) has additive complexity  k 1f lIud
only if there is a sequence of nonzero polynomials gl, . . ., 9k E JR[X] such that
(i)
91 = a1X"1 + b1 XI \
with al, b l ER, al = (au,..., a1n) E N n , flt = (/111,.' .,/11n)E N n ,
(ii) for 1 < j  k,
j-1 j-1
gj = ajX"; II gt;; + bjXfJ; II g? ,
1=1 1=1
with aj,bjEJR, aj = (aj1,...,ajn)eNn, /1j = (f3 j l>...,[3jn)EN n , dji E
N, ej;EN for 1  i < j, and

TRIVIALIZATION
153
(iii )
k k
I. X"Y II i? = CX6 II 9;;
jl j=l
with cEIR., I = hb""ln)EN n , 6 = (6 1 ,...,6 n )EN n , andr1,...,rk,
Sl," .,skEN.
PROOF. If such a sequence 91, . . ., 9k exists, then I is clearly of additive complexity
 k. Suppose I is of additive complexity  k, and let it,. . ., Ik be a sequence of
nonzero rational functions as in (3.4). By writing the integer exponents in the
expressions in (3.4) as differences of natural numbers and clearing denominators
one obtains a sequence 91,..., 9k of nonzefO polynomials in JR[X] satisfying (i) and
(ii) such that
/ j-I
fj = 9j X<; II gf;;
.=1
for 1  j  k, with Ej = (Ej1," .,Ejn)ENn and Pji EN fOf 1  i < j. Then f has
the form described in (iii). 0
(3.7) LEMMA. Suppose f E R[XJ has additive complexity  k. Let Y = (Yi,..., Y k )
be a tuple of new variables. Then there are polynomialsht(X,Y),...,hk(X,Y)E
JR[X, Y] such that each h j has at most three monomials, and there are two mono-
mials J.l1 (X, Y) and J.l2(X, Y) and a real constant c such that:
(i) lor each XEJR n there is a unique yEJR k with h 1 (x,y) =... = hk(x,y) = 0;
(ii) lor each x E JRn, ily E JRk and ht(x,y) = ... = hk(x,y) = 0, then
J.l1(x,y)f(x) = CJ.l2(X,y);
(iii) the set
{XEJR n : if YER k and hl(x,y) = '" = hk(x,y) = 0, then J.lt(x,y) #- o}
is open and dense in JR n.
PROOF. Take polynomials 91,... ,9k E JR[X] as in the pfevious lemma. Then we
put
ht ,- Yt - (a I X"l + b I X,81),
and for] < j  k, h j .- 1'; - ( ajx"; IT y;d;; + bjX{J; IT y;e;; ) .
;=1 ;=1
Also, we put III (X, Y) := X"Y rr7=, Y/; and J.l2(X, Y) := X 6 rr7=t Y/; with
"y, 0, rl,. . ., rk and Sl,.. ., Sk as in the previous lemma. Clearly
hl(x,y) =... = hk(x,y) = 0  yj = 9j(x) for j = 1,.. .,k,

154
TRIVIALIZATION
from which (i) and (ii) follow immediately. Since gl, . . ., gk are nonzero polynomials
we also get property (iii). 0
(3.8) To see how the last lemma is to be applied, note that conditions (i), (ii) <tlld
(iii) definably determine the function x ...... I(x) :JR fi ..... JR in terms of the polynomi;;ls
hi,. .., h k , the monomials J.Ll and J.L2, and the constant c: it is the unique continuol)s
extension to JRfi of the function x H CJ.L2(X,Y)/J.Ll(X,y) defined on the open dl'IISP
set in (iii), where yERk is given by h 1 (x, y) = '" = hk(x, y) = O.
(3.9) EXERCISE. Show that there is no function F:JR 2 .....JR, definable with pararll-
eters in Roxp, such tat F(x,k) = xk for all XEJR and kEN.
Hint: use that R. xp is o-minimal.
Notes and comments
The semialgebraic triviallzation theorem over JR is due to Hardt [29], who noll's
that his arguments go through for the category of bounded subanalytic sets. For
similar results about the category of Nash manifolds, see Coste and Shiota [10J.
Here we follow closely the treatment of the semialgebraic case over arbitrary real
closed fields in Delfs and Knebusch [11]. There is a difficulty similar to the olle
in the previous chapter. This explains why the key lemma (1.5) was obtainpd
in a somewhat roundabout way (first doing the bounded case, and then reducill/';
to that case) compared to the route available for the corresponding semialgebraic
result (6.3) in [11].
Subsequently (in 1993) I found the easy theorem (2.2) of Chapter 6, which IE'ads to
the short model-theoretic proof in Chapter 8, (3.13) and (2.14), that triangul<ttion
implies trivialization. But it seemed reasonable to keep also the longer but qllitp
explicit "geometric" construction of the trivialization. The local conical strllrtllfC'
of definable sets (theorem (2.3)) is a familiar consequence oftrivialization, as in tlrC'
semialgebraic case treated in [4].
As soon as Wilkie's theorem was available (in 1991) it was clear that one could
use the trivialization theorem with distinguished subsets to answer the question of
Risler and Benedetti as done in Section 3.
The following came too late to my attention to include in the references at the l'lId
of the book. It is closely related to the material of this chapter.
M. Coste, Topological types of feumomials, preprint, December 1996.

CHAPTER 10
DEFINABLE SPACES AND QUOTIENTS
Introduction
1:p till now our definable sets were always given as subsets of an ambient space
R m , a very convenient restriction that has served us well. In this final chapter
we want to break out of this restricted setting, and consider also, say, projective
space, and its "definable" subspaces, more generally, spaces that are not given as
su bsets of Rm, but locally look like definable subsets of Rm. To stay within the
context of "spaces of finite type" we require a covering of the spaces of interest by
only finitely many "affine" definable patches. This idea is carried out in detail in
Section 1. The main result of this section, theorem (1.8), generalizes a theorem of
Robson for semialgebraic spaces, to the effect that a "definable space" (obtained
by gluing finitely many affine definable sets) is isomorphic to an affine definable set
if and only if the space is regular, a separation condition that is easily verified in
many situations of interest.
In Section 2 we consider a related construction, namely that of taking the quotient
space X / E of a definable set X by a definable equivalence relation E on X. (De-
finably gluing finitely many definable sets can be viewed as a special case of this
construction, but is better treated separately, as We do in Section 1.) We ask when
X / E can be realized as an ordinary definable set, and are particularly interested
in the case that E is "definably proper" over X. The main positive result in this
direction extends a semialgebraic theorem of Brumfiel to our setting. The proof
follows Brumfiel's, with some modifications. Triangulation is the key tool.
The two sections are independent. The point of this chapter is that the entire
theory of the previous chapters is not restricted to just definable sets in R m : we
are free to use certain constructions on these definable sets that carry us outside
Rm.
We fix an o-minimal expansion 1l = (R, <, . . .) of an ordered real closed field.
"Definable set" will always mean "definable subset of R m for some m", unless
specified otherwise.
155

156
DEFINABLE SPACES AND QUOTIE:-ITS
1. Definable spaces
(1.1) We start by recalling the construction of a topological space by gluing:
Let a covering S = Ui U, of a set S by subsets U, (i E 1) be given, and for eil<.h
index iE/ a set-theoretic bijection gj : Ui - U:, with U: a topological space, stlrh
that for all i,j the set g,(Ui n U j ) is open in U: and the "transition" map
gij :gi(Ui n Uj)-gj(Uj n U,) with gij(a:) = gj(gi-l(a:»)
is continuous (hence a homeomorphism, since Uij is a bijection with inverse 9..;).
Then we equip S with the unique topology in which each Ui is open, and earh
gi : Ui - U[ is a homeomorphism. (In this topol<;lgy a subset X of S is opel] iff
gi(Xn Ui) is open for all iE/.)
(1.2) Suppose in addition that the index set / is finite, that each U:  Rm(i) j,
a definable set (with the induced topology from Rm(i», and that for each pair i,j
the set gi(Ui n Uj)  U[ is definable and gij:gi(Ui n Uj)-gj(Uj n Ui) is definah!e.
(Such a family (g, : U, - uI) iEl is called a definable atlas on S, and the 9,'s
charts of the atlas.)
Then we extend the notion of "definable set" to subsets of S: Let X  Sand
f : X -+ R; call X definable if gi( X n Ui)  U: is definable for each i, and cali f
definable if X is definable and fi: gi(X n Ui) - R given by fila:) = f(g;l (a:» is
definable for each i. (We need this mainly for open X and continuous f.) Note
that the definable subsets of S form a boolean algebra of subsets of S, and that
for given definable X  S the definable functions f : X -+ R form an R-algebra
under pointwise addition and multiplication of functions. (The constant functions
X -+ R are of course definable.) If X = Xl U ... U Xn, where all X k are definabll',
then a function f: X - R is definable iff each restriction flXk :X k -+ R is definable.
Kote that U i is definable, and for gi = (gil,...,gim(i») : Ui - Rm(i), each 9;, is
definable. Let DO(S) be the collection of definable open subsets of S, and for eitch
U E DO( S), let DC( U) be the R-algebra of definable continuous functions f : U  H.
Then we call the set S equipped with the sheaf (DC(U»)UEPO(S) a definable space.
This is of course only a sheaf in a finite sense: if U = Ul U.. 'UU n where Uk E DO(S)
for all k, then a function f: U -+ R belongs to DC( U) iff flUk : Uk -+ R belon!1: s
to DqUk) for all k. The topology on S is not lost in viewing S as a def1ll3 ble
space, since DO(S) is a basis for the topology. However, the particular definab[('
atlas (g; : U i - Un iEl that makes S into a definable space is not recoverahlc
from the sheaf, and different definable atlases on S can make S into the sainI'
definable space. Two definable atlases (gi : Ui - Un iEl and (hj : Vj - vj) j(.1
(on the abstract set S) are called equivalent, if for all iE/ and j E J the se:"
gi( U i n v:,) and gj(Vj n U i ) are open definable subsets of U[ and VJ respectively, alld
the transition maps a: ...... hj(gjl(x») :g,(Ui n Vj) _ hj(Vj n U i ) and their inverses
y ...... gj(hjl(y») are definable and continuous. One checks easily that then (gi) and
(hj) give rise to the same topology on S, to the same notion of definable subset of
S, and to the same notion of definable function on a definable subset of S.

DEFINABLE SPACES AND QUOTIENTS
157
COSCLUSION. Two definable atlases (9i) and (h j ) on S are equivalent iff they give
rise to the same sheaf on S, that is, they make S into the same definable space.
The notions of definable subset of S and definable function on a definable subset
of S are now seen to depend only on S viewed as a definable space, not on the
particular definable atlas used to make S into a definable space.
(1.3) Let T be a second definable space with sheaf (DC(V») VEPO(T)' Then a
morphism F : S -+ T is a map from S into T, such that if V E DO(T) and
hE DC(V), then F-l (V) E DO(S) and h 0 (FIF-l (V») E DC(F-l (V)). It follows
in particular that F is continuous. If the definable space T is determined by the
definable atlas (h j : Vj -+ Vj) jEJ' then a map F : S -+ T is a morphism iff for
all i E I and j E J the set 9i(Ui n F-l(V j ») is definable and pen in U:, and
the map 9i(U i n F-l(Vj)) -+ VJ given by x ,..... hj(F(g;l(x»)) is continuous and
definable. Let F: S -+ T be a morphism. Then, given any definable set Y S;; T
and definable function f: Y -+ R, the set F- l (Y) S;; S is definable and the function
f 0 (FIF- l (Y)) : F- l (Y) -+ R is definable. Also, if X S;; S is definable, then
F(X) S;; T is definable.
The identity map S -+ S is a morphism, and each constant map S -+ T is a
morphism. The definable spaces with their morphisms form a category under the
usual composition of maps.
(1.4) EXAMPLES.
(1) Each definable set S S;; R m is made into a definable space by taking the identity
map S -+ S as the only chart of a definable atlas. Then the topology of the
definable space S equals the topology induced by Rm, and set X S;; S is definable
in the definable space S iff it is definable as a subset of R m , and in that case a
function f: X -+ R is definable in the sense of the definable space S iff r(f) is a
definable subset of R m +1. If T S;; R n is a second definable set, then a morphism
S -+ l' is the same thing as a continuous definable map S -+ 1'. An affine definable
space is by definition a definable space isomorphic to a definable set (in the category
of definable spaces with their morphisms).
(2) (Projective spaces) Let pn(R) be n-dimensional projective space over the field Rj
its points are the equivalence classes (xo : ... : x n ) of nonzero vectors (xo,..., x n )
in R n +1, where two such vectors are equivalent iff one is a scalar multiple of the
other. Clearly pn( R) is covered by the n + 1 subsets Ui (0 :$ i :$ n), where Ui is the
set of points (xo : " . : Xi : . .. : x n ) with Xi #- 0 (or equivalently, with Xi = 1). Let
gi : U j -+ R n be the bijection (xo : ...: 1 :...: x n ),..... (xo,.. "Xi-l,xHl,.. .,x n ).
It is easy to see that (gi)O<i<n is a definable atlas, and we make pn(R) into a
definable space using this atlas. In fact, pn(R) is an affine definable space. To see
this, define v: pn(R) -+ R(n+1)' by
V(XO: Xl:"': X n ) = (XiXj/(X +...+ x))o:Si.j:sn'
Then v is easily seen to be an isomorphism from the definable space pn(R) onto a

158
DEFINABLE SPACES AND Ql:OTIENTS
definable subset of R(n+1)' .
(3) (Subspaces) Let S be a definable space given by a definable atlas (9i : t', -.,
Un El , and let X  S be definable. Then (gilUi n X: U i n X....... gi(Ui n X))i(:[ i, it
definable atlas on the set X, and we make X into a definable space using this at;:t,.
We call X in the role of definable space a definable subspace of S. The topolo1-\Y
of X as a definable space is the one induced by 5. The definable subsets of tit"
definable space X are exactly the definable subsets of 5 that are contained in X.
and given a definable set Y  X, a function f : Y .... R is definable in the Bt'f,'" of
the definable space X iff f is definable in the sense of S. (Thus another (equi\'a:..nt)
definable atlas on S makes X into the same definable space.) The inclusion X - . S
is a morphism. If F: S .... T is a morphism between definable spaces Sand T, and
F(S)  Y, where Y is a definable subset of T, then F is also a morphism wlt"n
considered as a map from 5 into the definable subspace Y of T.
(4) (Products) Let 51,,," 5 K be definable spaces, and take for each Sk a definit:,;e
atlas (9ki: Uki ....... UJ,i)El., Uki  Rm(Jci), 1  k  K.
Put 1:= 1 1 x... X I K , and for each i = (i 1 ,.. .,iK)EI, let
Ui .- U 1i , X"'XUKiK 51 X"'X5K,
U: := U: i , X ... X U KiK  R m (li,)+..+m(Ki K J,
gi := g1i, X . . . X gKiK : Ui ....... U:.
Then (gi : U i .... U:)iEl is a definable atlas on Sl X ... X SK and we make 8 1 x
. .. X 5K into a definable space using this atlas. This gives the product topolOKI' on
Sl X '" X SI(' Moreover, given a definable space X and morphisms J.lk : X  Sk
for k = 1,.. .,K, the map /-I = (/-11,.. .,J.lK): X.... 51 X... X 5K is a morphism. It
follows easily that the definable space structure on the product set 51 X. . . x SJ( do('
not depend on our choice of definable atlases for the definable spaces 51,' . . , 81';'
(5) (Definable maps) Let S and T be definable spaces. Then a map F: S.... T is said
to be definable if its graph f(F) is a definable subset of the product space S x 'J',
as specified in (4). One checks easily that for T = R this agrees with the function
F: S....... R being definable in the sense of (1.2). Also F: S....... T is a morphism iff F
is definable and continuous.
(1.5) DEFINITION. Recall that a topological space S is said to be regular if for
each a E S and open U  S with a E U there is an open V  5 with a E V and
cl(V) £ U. We leave it as an exercise to show that a definable space S is regular if
and only if for each a E S and definable open U  S with a E U there is a defina hk
open V  S with a E V and cl(V)  U, and also if and only if for each a E Sand
definable closed X  5 with aX there are disjoint definable open neighborho()':s
of a and X in 5.
Note that each affine definable space is regular. The theorem below states tit,.
converse. A regular topological space is clearly Hausdorff, but the following is an

DEFINABLE SPACES AND Ql;OTIENTS
159
eXample due to Robson of a definable Hausdorff space S that is not regular, and
hence not affine.
EXAMPLE. In the o-minimalstructure (R,<,O,I,+,-,.),let Ul = U{ ]Rz be the
union of the open unit disc and the point p := (0,1). Let Uz = U  ]Rz be the
open square {(x,y) : 0 < x < 1 and 0 < y < I}. Let S := Ul U Uz as a point set
and consider S as a definable space via the identity mappings gi : Ui -- U: for
i = 1,2. With these definitions, S is a definable Hausdorff space. Now consider the
arc"y := {(x, y) : XZ + yZ = 1, x > 0, Y > O}  U z : "'I is closed in S since it is closed
in Uz and disjoint from U 1 , while any open subset of Uz containing "'I meets Ul in
an open set whose closure contains p. Thus p and "y cannot be separated.
(1.6) DEFINITION. Let S be a definable Hausdorff space and (a,b) an interval. A
definable map "y : (a, b) -- S is called a definable curve in S. For XES, we write
"y -+ x to mean limj....b "'I(t) = x. We call "'I completable if there is a (necessarily
unique) point xES such that "y -+ x. If "'I -+ x E X with definable X  S such
that "'I ( a, b)  X, we call "y completable in X. With these definitions one easily
verifies the following facts using (4.1) and (4.2) of Chapter 6:
(1.7) LEMMA. Let Sand T be definable Hausdorff spaces, f : S -+ T a definable
map, and xES. Then f is continuous at the point x if and only if for each definable
curve"y in S with, -+ X we have f(,) -+ f(x), where fb):= f 0 ,.
(1.8) THEOREM. Every regular definable space is affine.
PROOF. Let S be a regular definable space given by the definable atlas (hi: Ui --
Vi) iEl with definable Vi  Rn,. We clearly can assume each Vi is bounded. Let
I = IJI. The case I = 1 is trivial. If we assume that the theorem holds for I = 2,
then a straightforward argument proves it for all I.
We now turn to the proof ofthe case 1= 2; say I = {1,2}. If X  R n , we denote
by OX the set cI(X) - X. Writing Vu := hl(U1 n Uz) and V21 := hz(Uz n Ud, the
following definable sets need to be considered:
Bl := VI n OViz = hi (aUz),
Bz := Vz n OV 21 = hZ(aUl),
B; := {x E R n l : 3y E B Z V£I' £z >0 3zE U 1 n Uz (d(x, hl(z)) < £1, d(y, hz(z)) < £2j},
B; := {YE R n . : 3x E Bl V£l, £z > 0 3z E U 1 n U z [d(x, h 1 (z)) < (1, d(y, hz(z)) < (Ij}.
The following fact about these sets is crucial to the construction:
CLAIM 1. d(x,BD>OjoreveryxinV;,i=1,2.

160
DEFINABLE SPACES AND QUOTIENTS
PROOF. We prove the claim for i = 1, the case i = 2 being symmetrical. Assulnr
for a contradiction that d(x,BD = 0 for some x E VI. Note that h 1 1 (x)EU 1 and
U 1 n h;I(Bz) = 0, so UI n clh;I(B z ) = 0. Since S is regular there are disjoint
open neighborhoods D of h11(x) in UI and E of h;I(B z ) in Uz. We will deri\'(,
a contradiction by finding an element in D n E. Since hI (D) is open in VI. WP
can take £ > 0 such that B(x, £) n VI  hI (D). Because d(x, BD = 0 tber( is
x' E Bi with d(x, x') < £. By definition of Bi there is y E Bz for which there are
points ZEU 1 n U z with hl(z) arbitrarily close to x' and hz(z) arbitrarily close to y.
Since hz(E) is an open neighborhood of y in V z it follows in particular that there
is z E UI n Uz with d(x, hl(z» < £ and hz(z) Ehz(E), hence zE D n E. This prow's
the claim. 0
Let dl(z) := d(hl(z), BD and dz(z) := d(hz(z), BD for z E S. We now define tllP
map h: S..... RI+n,+1+n. as follows:
{ (dl(Z), dl(z)hl(Z), 0 ,0,..., 0) for zE U 1 - U z ,
h(z):= (dl(z),dl(z)hl(Z), dz(z),dz(z)hz(z») for zE UI n Uz,
( 0 ,0,... ,0, dz(z),dz(z)hz(z») for zE U z - U 1 .
By claim 1 we clearly have for each z E S
h(z)EUI - Uz <=> the last 1 + nz coordinates of h(z) are equal to 0,
h(z) E Uz - UI <=> the first 1 + nl coordinates of h(z) are equal to O.
From this it follows easily, again using claim 1, that h is injective.
CLAIM 2. The map h is continuous.
To see this, let "y be a definable curve in S with "y ..... z E S. We have to show that
hb) ..... h( z). By restricting the domain of"Y suitably we may assume that I lies
either completely in U 1 - Uz, or completely in U 1 n U z , or completely in U z .- C I .
CASE 1. "Y lies completely in UI-UZ' Then also zE U1-Uz, since zE U z would imply
that "Y lies at least partly in U z . The definition of h now gives that h( "Y) ..... h( z).
CASE 2. r lies completely in U I n U z . If also zE UI n U z , the definition of h gives
h(r)"'" h(z). So let z U 1 n U z , say z UI' Then zEf}U 1 , so y := hz(z) E Bz. Tahl'
x E R n , such that hi ("Y) ..... x. Since h z ("'f) ..... Y E Bz it follows that x E Bi, hen ce
hC"Y)..... h(z) as the formula above shows.
The case that "'flies completely in U z - U 1 is symmetric to case 1. This finishes the
proof of claim 2. 0

DEFINABLE SPACES AND QUOTIENTS
161
CLAIM 3. h maps S homeomorphically onto h(S).
To prove this, let K be an arbitrary definable closed subset K of S. In view ofthe
injectivity of h and claim 2 it suffices to show that h(K) is then closed in h(S). Let
z E S with h(z) E cI(h(K)) n h(S): it is enough that we derive from this that z E K.
Since h(z)EcI(h(K)) there is a definable curve "Y in K such that h("Y) -+ h(z). We
may assume that "Y lies either completely in Ul - U 2 , or completely in Ul n U2, or
completely in U 2 - U 1 . We may also assume the domain of, is an interval (0, c).
CASE 1. The curve "Y lies completely in Ul - U2. Since
h(,(t)) = (d 1 (,(t)),d 1 ("Y(t))hl("Y(t)),0,0,...,0) -+ h(z) as t -+ €,
the last 1 + n2 coordinates of h(z) are 0, so that z E Ul - U2, and
h(z) = (d 1 (z),d 1 (z)h 1 (z),0,0,...,0).
Using claim 1 this gives hl(,) - h 1 (z), hence "Y -+ z, so zEK, as K is closed in S.
CASE 2. The curve "Y lies completely in U 1 n U2. Then h("Y(t)) -+ h(z) as t --+ €,
where
h ("Y(t)) = (d 1 ("Y( t)), d 1 (,(t)) ht (,(t)), d2 ("Y(t)).d2 (,(t))h2 ("Y(t))).
If z E Ul, then h(z) = (dl(z), d l (z)h 1 (z),...), so that by claim 1 we obtain ht ("Y) -
h 1 (z), and thus, -+ z (since h 1 is a homeomorphism onto Vi), hence z E K. If
z E U 2 the argument uses instead the last n2 coordinates of h(z).
The case that ,lies completely in Uz - U t is symmetric to case 1. This finishes the
proof of claim 3, and thereby the proof of the theorem. 0
2. Definable quotient spaces
(2.1) Given a definable map I:X -Y between definable sets X and Y, we denote
by Ej the kernel of I, that is, Ej = {(x,y)EX X X: I(x) = I(Y)}, a definable
equivalence relation on the set X. Note that if I is continuous, then Ej is closed
in XxX.
(2.2) Let E  X X X be a definable equivalence relation on a definable set X  Rm.
DEFINITION. A definable quotient of X by E is a pair (p, Y) consisting of a
definable set Y  Rn and a definable continuous surjective map p : X -+ Y such
that:
(i) E = Ep, that is (Xl, X2) E E <=> p(Xt) = p(xz), for all Xl, Xz E X;
(ii) p is definably identifying, that is, for all definable K  Y, if p-l(K) is
closed in X, then K is closed in Y. (See Chapter 6, (4.4).)

162
DEFINABLE SPACES AND QUOTIENTS
(2.3) REMARKS.
(1) If clause (ii) above is replaced by the stronger clause
"p is definably proper",
then we call (p, Y) a definably proper quotient of X by E. (In th
presence of surjectivity "definably proper" implies "definably identifying",
see Chapter 6, (4.6).)
(2) If (p, Y) and (p', Y') are both definable quotients of X by E, then by (i) we
clearly have a unique definable bijection h: Y -+ Y' such that the diagram
X
;/
Y
h
. Y'
commutes; by (ii) this bijection is a homeomorphism.
Loosely speaking this means that up to isomorphism there is at most olle
definable quotient of X by E. If such a quotient (p, Y) exists, we are
therefore justified in calling it the definable quotient of X by E, and write
Y = X IE. We shall also use the phrase "p: X -+ Y is a definable quotien t
of X by E" instead of "(p, Y) is a definable quotient of X by E".
(3) A definable quotient p: X -+ Y is a quotient of X by E in the category of
definable sets and continuous definable maps, in the sense that if f:X -. Z
is any continuous definable map into a definable set Z  R n such tr.at
E  E J, then the unique map g: Y -+ Z such that f = 9 0 P is conti!luolls
and (obviously) definable.
(4) For a definable quotient of X by E to exist it is clearly necessary that E is
closed in X x X. But this is by no means sufficient, see Brumfiel [7, p. 71].
(2.4) EXAMPLE. Let X  R m be definable and A  X a nonempty definable
subset of X. We would like to "definably collapse" A to a point. In precise terms,
consider the definable equivalence relation E A on X whose equivalence classes <in'
the singletons {x} with x E X - A, and the set Aj we would like to find the definabl
quotient XI EA' Concerning this we have the following result.
PROPOSITION. If A is closed and bounded in Rm, then there is a definable proper
quotient of X by EA'
PROOF. Consider the definable continuous map p: X -+ Rm+1 given by p(:c) .
(dA(x). x,dA(x»), where dA(x) := inf{lx - al: a E A} is the distance from x to
A. Put Y := p(X), and consider p as a map X -+ Y. Assume A is closed and
bounded. Claim: (p, Y) is a definably proper quotient of X by EA. Clause (i) is
clearly satisfied. To check the "proper" version of (ii), consider a definable curve
'Y: (b, c) -+ X such that p('Y) -+ p(x) E Y, x E X. By Chapter 6, (4.5) it suffices to

DEFINABLE SPACES AND QUOTIENTS
163
show that 'Y is completable in X. If limt-+c d A b(t») = 0, then 'Y -+ a for some a E A,
since A is closed and bounded in R m , and we are done. So we may assume that
IimHc d A bet») == dA(x) 1: O. In combination with limt-oc d A (-r(t») ''Y(t) = dA(x).x,
this gives limt-+c I(t) = x, and we are done again. 0
This is a very special case of our main theorem (2.15), but before we can get to that
we need a number of preliminaries, on completions, disjoint sums, and attaching
spaces respectively.
(2.5) COMPLETIONS.
A completion of a definable set S  R m is a pair (h, S') consisting of a closed
and bounded definable set S'  R" (for some n) and a definable map h: S -; S'
such that h is a homeomorphism from S onto h( S) and h( S) is dense in S'. We
also express this more informally by: h: S -+ S' is a completion of S.
Clearly each definable set S  R m has a completion: just take a definable map
J.t: R m -+ R m that maps R m homeomorphically onto (-1, l)m, put S' := closure of
J1.(S), and let h:S-+S ' be the restriction of J1.. Then h:S.....S ' is a completion of S.
(2.6) Let f: S..... T be a definable continuous map between definable sets S  R m
and T  Rn. Then a completion of f: S..... T is a triple consisting of a completion
i : S..... S' of S, a completion j: l' -+ T', and a definable continuous map f': S' .....1"
such that
f' 0 i = j 0 f.
We also express this by saying that the commutative diagram
S i S'
----+
(*) 11 if'
T ----+ T'
j
is a completion diagram of f:S-+T.
There is always a completion of f: S -+ T, but for our purpose we need a more
precise result:
(2.7) LEMMA. Let a completion j: T -; T' be given. Then there is a completion
diagram (*) of f: S ..... l' with the given map j as the bottom map in (*). If in
addition S is given as a definable closed subset of a definable set X  R m , then
there are a completion h : X ..... X' and a completion diagram (*) of f, such that
S' = closure of h(S) in X', and i = h[S:S-+ S'.
PROOF. Choose a completion g : S -+ SC of S, and note that the definable map
g X j : S x T -; SC x T ' is a homeomorphism onto its image g(S) x jeT). Let

164
DEFINABLE SPACES AND QL"OTIENTS
S' be the closure of (g x j )(f(J») in SC x T'. Then i: S --+ S' given by i( x) '"
(g x j)(x,f(x») = (g(x),jf(x») is a completion of S, and the restriction of thr
projection map SC x 1" --+ 1" to S' is a definable continuous map f' : S' --+ T'
making the diagram (*) commutative.
Suppose now that S is a definable closed subset of the definable set X  R m . In tfI('
ambient affine space of 1" we take a closed ball B that contains T'. By Chapter 1<.
(3.10) there is a definable continuous extension j:X --+B of j 0 f: S --+ 1". Apply tl:t'
first part of the lemma to get a completion of j: X --+ B consisting of a completion
h: X --+ X', the completion 18: B --+ B, and a definable continuous map jr : X'  1/
such that jr 0 h = j. Put S':= closure of h(S) in X', i:= hIS:S--+S', and not('
that jr(S')  T'. Hence f' := ]'IS' provides us with a completion diagram (*) a,;
required. 0
(2.8) DISJOJ:iT SeMS.
Let S1>''', Sk be definable sets in Rm(I),..., Rm(k), k  1.
DEFINITION. A disjoint sum of SI,.", Sk is a tuple (hI,"" h k , 1') consisting or
a definable set l'  R" for some n, and definable maps hi: Si --+ T such that:
(i) hi is a homeomorphism onto hi(Si) and hi(Si) is open in 1', for i = 1,.. ., A';
(Ii) Tis the disjoint union ofthe sets h l (SI)" ..,hk(Sk)'
Let n = 1 + max{ m(i): 1:5 i :5 k} and define hi : Si--+ R" by h;(x) := (x, i,..., i).
Then (hI,"" h k , Ui hi(Si») is clearly a disjoint sum of SI,.. ., Sk.
If (h1>...,hk,1') and (hL...,h,1") are both disjoint sums of SI,,,,,Sk, thrn
there is a unique definable homeomorphism  : l' --+ 1" such that  0 hi = Ii:
for i = 1,..., k. This means, loosely speaking, that there is up to isomorphism
exactly one disjoint sum of SI,' . ., Sk. In the following we fix such a disjoint sum
(h1>"" hk, T) of SI,.. ., Sk, and we write SI II. ..II Sk for 1'j there will be no harlll
in identifying each S; with its image in SI II . .. II Sk via hi'
(2.9) CO:iSTRUCTIO:i OF X II! Y.
Let X  R m and Y  R" be definable sets, and let a definable continuous map
f : A --+ Y from a definable set A  X into Y be given. We would like to Mtach
X to Y via f. To this end we consider on the disjoint sum X II Y the smallest
equivalence relation E(J) for which each aEA  X is equivalent to f(a)EY. One
checks easily that
E(f) =(X)U(Y)U{(a,f(a»): aEA}U{(J(a),a): aEA}
U {(al,az)EA x A: f(ad = f(az)},
where (X) = {(x,x): XEX} and (Y) = {(y,y): yEY} are the diagonals of
X and Y. This shows that EU) is a definable equivalence relation; the question is

DEFINABLE SPACES AKD QL"OTIENTS
165
whether the definable quotient of X II Y by E(f) exists. If it does, we denote this
quotient by X III Y rather than by X II YI E(f), and we say that X III Y is the
space obtained by attaching X to Y via f.
(2.10) U;\"[VERSAL PROPERTY OF X lIlY'
Suppose X II I Y exists. Let : X -+ X III Y be the composition
X -+ XIIY -+ XIII Y,
and similarly for 1/: Y -+ X II I Y. Clearly  and 1/ are definable and continuous and
(a) = 1/(J(a)) for all a E A. Moreover, given any two definable continuous maps
e: X -+ Z and 1/': Y -+ Z into a definable set Z with '(a) = 1/'(J(a)) for all a E A,
there is exactly one map /-!: X III Y -+ Z such that /-! 0  = e and /-! 0 1/ = 1/'; this
map /-! is continuous and (obviously) definable.
(2.11) LEMMA. Let A be closed and bounded in the ambient space R m of X. Then
X III Y exists as a definably proper quotient of X II Y.
PROOF. If A = 0, then the identity map X II Y -+ X II Y is clearly a definably
proper quotient of X II Y by E(f). So let us assume A 1: 0. Let R M be the ambient
space of X II Y. We identify X and Y with their images in X II Y; note that A is
then closed and bounded in R M . Let d A : RM -+ R be the distance function given
by
dA(z):= min{lz - al: aEA}.
Let also 1: X -+ R M be a continuous definable extension of f: A -+ Y. (Such j
exists by Chapter 8, (3.10).) Define the map p:X II Y -+R 2M +1 by
p(x) = (l(x),dA(x). x,dA(x)) for xEX and p(y) = (y,O,O) for y E Y,
so p is definable and continuous, and once checks easily that Ep = E(f). Let
Z = p(X II V), a subset of R 2M +1. We will show that (p, Z) is the desired definably
proper quotient of X II Y by E(f). Let "t be a definable curve "t in X II Y such
that p( "t) is completable in Z.
By Chapter 6, (4.5) it suffices to show that "t is completable in X II Y. We may as
well assume (after restricting the domain of "y suitably) that either "t lies entirely
in X, or "t lies entirely in Y. In the first case we may further assume "t lies entirely
in X - A and is bounded away from A since otherwise "y will be completable in X
with "t -+ a E Ai with this further assumption on "y and using the assumption that
p( "y) is completable in Z it follows just as in the proof of (2.4) that "t is completable
in X. The case that "y lies entirely in Y is even simpler and left to the reader. 0
We now use completions to extend this lemma as follows:

166
DEFINABLE SPACES AND QUOTIENTS
(2.12) PROPOSITION. Suppose A is closed in X and f:A-+Y is definably proptT.
Then X II J Y exists as a definably proper quotient of X II Y.
PROOF. Identifying X and Y with their images in suitable completions and USiIlf!;
lemma (2.7) we may assume that X and Yare bounded in their ambient Spa«'
and f extends to a definable continuous map cl(f) : cl(A) -+ cl(Y). Since f is
definably proper we have cI(f)-1(y) = A. (To see this, let x EcI(A), cI(f)(x) E 1".
To derive x E A we take a definable path "'I : [0,1] -+ cI(A) with "'1(1) = x and
"'I GO, 1))  A. Then cl(f) (, ([0, IJ))  Y is closed and bounded in the amhient
affine space of Y, so f-1 (clU) (-y ([0,1]))) is a subset of A and is closed and bound!',;
in the ambient affine space of X. Since this set contains "'1([0,1]) it must also
contain "'1(1) = x, hence x E A.) By (2.11) we have a definably proper quotient
cI(p):cI(X) II cI(Y)-+X II c 1(J) Y. Consider X II Y as a subset of cI(X) llcl(Y) in
the obvious way and let Z := cl(p)(XIIY). It is easily seen that cI(p)-1(Z) = X IJ}'
and that E (cI(J») n (X II y)Z = E(f). Hence p := cI(p)IX II Y: X II Y -+ Z is a
definably proper quotient of X II Y by E(f). 0
(2.13) Let E be a definable equivalence relation on a definable set X. Let PTJ :
E -+ X and prz : E -+ X be the restrictions of the two projection maps X x X -+ X.
We call E definably proper over X if prl (equivalently, prz) is a definably prop!'I'
map. One checks that if XI E exists as a definably proper quotient of X by E, then
E is definably proper over X. The main theorem (2.15) below asserts the convers('.
We need one more lemma before embarking on its proof.
(2.14) LEMMA. Let E  XxX be a definable equivalence relation on a definable sd
X  R m and suppose E is definably proper over X. Then, given a definable CUTT'
"'I = (a, /3) in E with a = prl ("'I), /3 = przb), if either a or 13 is completablc in X.
both are completable in X, and in that case, if a -+ p, /3 -+ q, then "'I -+ (p, q) E L.
(Hence E is closed in X X X.)
PROOF. Suppose a is completable in X. Since pr1 :E-+X is definably proper and
"y lifts a, it follows that "y is completable in E. Hence 13 = przb) is completabie in
X. 0
(2.15) THEOREM. Suppose the definable equivalence relation E on the definablr
set X is definably proper over X. Then XI E exists as a definably proper quoticnt
afX.
This contains proposition (2.12) on X IIJ Y as a special case, but (2.12) is needed
in the proof of (2.15). Here are two further important special cases:
(2.16) COROLLARY. If X  R m is closed and bounded and E  X X X is a closed
definable equivalence relation, then XI E exists as a definably proper quotient of X.

DEFINABLE SPACES AND Q'COTIENTS
167
(2.17) ApPLICATION TO ORBIT SPACES. Let G be a definable topological group,
that is, G is a definable subset of some Rm equipped with a definable continuous
group operation G X G -+ G. (We leave it as an exercise to show that then the group
inversion x ,..... x- 1 : G --+ G is also continuous, so that G is indeed a topological
group.) Let in addition a continuous definable group action G x X -+ X on a
definable set X be given. Then the orbits Gx are exactly the equivalence classes of
the definable equivalence relation "'G on X given by x "'G Y <=> gx = y for some
gEG.
(2.18) COROLLARY. If G is closed and bounded (for example, finite), then the
orbit space XIG := XI "'G exists as a definably proper quotient of X.
(2.19) PROOF OF THEOREM (2.15). By induction on dim(X). If dim(X)  0,
then X is finite; in that case the theorem holds trivially. Assume dim(X) = d > O.
By Chapter 6, (1.2), there is a definable set S  X that has exactly one point in
common with each equivalence class of E. Let (J : X -+ S be the definable map
that assigns to each x E X the unique point in S to which it is equivalent; (J may
not be continuous, let alone definably proper. Nevertheless we can use (J and S to
construct XI E. Let c( S) be the closure of S in X. Let B = (J (c( S) - S) be the
set of points of S that are equivalent to points of c(S) - S, so dim(B) < d. Take a
triangulation of X that is compatible with S, c(S) and B, let P be the partition of
X corresponding to this triangulation, and let Sd be the union of the d-dimensional
sets of P that are contained in S. One checks easily that Sd is open in X. Put
S' := c( S) - Sd. Since B n Sd = 0, no point of Sd is equivalent to a point of S'.
loreover S' is closed in X. Hence E' := En (S' X S') is definably proper over
S'. Since dimeS') < d there is by the inductive hypothesis a definably proper map
f' : S' -+ Y ' onto a definable set Y ' with E ' = Ef'. Note that f' maps S - Sd
bijectively onto V'. Let C(Sd) be the closure of Sd in X, and put A := C(Sd) n S'.
We construct Y = XI E by attaching C(Sd) to Y' via f" := I'IA:A-+ Y ' . Note that
A is closed in C(Sd), and that f" : A -+ Y ' is definably proper, since f' is. Hence
we can apply proposition (2.12) to obtain Y := C(Sd) II!" Y' as a definably proper
quotient of c( Sd) II y' via the map p: c( Sd) II Y' -+ Y.
f'ote that the composed map S' --+ Y' --+ Y agrees with the map C(Sd) --+ Y on
the intersection A of their domains, hence these two maps determine a (definable,
continuous) map 9 : c(S) = C(Sd) U S' -+ Y. Consider the following commuting
diagram:
C(Sd) II S'
j
-----+
C(Sd) II Y'
hl
lp
c(S) ---L., Y = C(Sd) IIJ" Y'
where the (continuous, definable) map j is the identity on C(Sd) and equal to I'
on S', and the (continuous, definable) map h is induced by the inclusion maps
C(Sd) -+ c(S) and S' --+ c(S). Note that all four maps are surjective, that p is
definably proper, and that j is definably proper since I' is. An easy diagram chase
then shows that 9 is definably proper. Since f' maps S - Sd bijectively onto y I ,

168
DEFINABLE SPACES AND QUOTIENTS
the map giS: S -+ Y is a bijection. Now put 1 := go u: X -+ S -+ Y. Clearly f j,
definable, surjective, and E = EJ. As mentioned already, u may not be continuous,
but we claim
(*)
f:X -+ Y is continuous and definably proper.
(Clearly (*) implies that I:X -+Y is a definably proper quotient of X by E, so "('
have reduced to proving (*).) To check continuity of I, take a definable curve n in
X with a -+ pEX. Then (a,u(a)) is a definable curve in E and a is completablp
in X, hence O"(a) is completable in X by lemma (2.14), say O"(a) -+ qEc(S). TllP/I
(p,q) E E, and I(a) = g(u(a)) -+ g(q). But g(q) = I(q): if q E S, this follows
from u(q) = qi if q E c(S) - S, then u(q) E S - Sd. and both q and u(q) are in S',
so that f'(q) = I' (O"(q)) , hence g(q) = g(u(q)) = I(q). Also I(q) = I(p). Ilel\('
I(a) -+ I(p). This proves continuity of f. To show 1 is definably proper, takE' a
definable curve a in X such that 1(0:) = 9 (O"( a)) is completable in Y. Sincc 9 is
definably proper, 0"(0:) is completable in c(S). Again by lemma (2.14) it follows
that a is completable in X. 0
Notes and comments
The notion of definable space (even for arbitrary o-minimal structures) is implicit
in Pillay [48]. It is used there to study the properties of groups and fields tli<tl
are definable in o-minimal structures. See also Khovanskii [33] for similar glui/lg
constructions in the Pfaffian setting.
For the proof of theorem (1.8) I largely followed Robson's treatment in [50] of tllP
semialgebraic case. The material on completions in Section 2 is adapted from Ddr,
and Knebusch [12], and the remainder of Section 2 follows Brumfiel [8].
For o-minimal expansions of "the real field with restricted analytic functions" thE'l'E'
is also a quite different way to go beyond the affine setting, by introducing a (locally
defined) notion of "nice" subset of an arbitrary real analytic manifolds, generalizi/ll!;
"sub analytic" sets; see [23].

HINTS AND SOLUTIONS
Chapter 2, (3.7)
3. Write Rm as a disjoint union of semialgebraic sets Ao, AI, 0 . . , Ak with AI, . . . , Ak
connected such that if x E Ao, then Q(x, T) = 0, while if 1  i  k and x E
Ai, then the real roots of Q(x, T) are given by continuous semialgebraic functions
(il(X) < ... < (ie(i)(X) of x. Note that then 9 must coincide on each Ai (i > 0)
with one of the functions (ij. Next use the fact that each point in Ao is in the
closure of Al U . . . U Ako
Chapter 3, (2.19)
3. Find a partition of G into cells G I ,..., G k and for each i = 1,..., k find contin-
uous definable functions liO,. . ., lir(i) :G i -+ R with
alGi = liO < Iii < . . . < lir(i) = .BIG;,
such thatforeachcell fC'Yij) (1  j < r(i» and each cell ({ij>/ij+d (0  j < r(i»:
(i) the restriction of f to f(")'ij) is continuous,
(ii) the restriction of f to ('Yij> 'YiHd is continuous, and either strictly increasing
in the (m + 1 )Ih variable, or independent of the (m + 1 )Ih variable, or strictly
decreasing in the (m + l)lh variable.
Then one shows along the lines ofthe proof oflemma (2.16) of Chapter 3 that each
restriction fiCiiO, 'Yir(i) is continuous.
Chapter 3, (3.8)
Given an L-formula q'>(XI, 0'" x m , y), there arc natural numbers M and N such
that for all r E R m the set r/J(r, 'R.)  R is a union of at most M intervals and at
most N points. This property is inherited by 'R.'.
Chapter 4, (1.17)
3. use the canonical homeomorphism of A with an open cell to show that a
non empty definable open subset of A has dimension d.
4. Partition A into finitely many cells, and note that {aEcI(A): dima(A) < d} is
contained in the union of the closures of the cells of dimension < d in this partition.
169

170
HINTS A:-ID SOLUTIO:-lS
Chapter 6, (1.15)
2 and 3. With T  X as in exercise 1, let I: X .....1' be the map assigning to ea('h
X its representative in T. Now apply (1.6)(ii) and (2.11) from Chapter 4.
4. If for a certain ( > 0 there is no such 5, find definable continuous maps 11, /2 :
(O,a].....X for some a > 0 such that 1/(!1(t») - 1(!2(t») I  (for all tE(O,a], alld
limf-tO I,l (t) - 'Y2( t): = o.
5. Consider points in X where the function x 1-+ If( x) - xl : X ..... R takes ih
minimum value.
Chapter 6, (4.8)
2. Suppose f is definably proper with respect to (R, <, S), and let (} be a curvp ill
X definable in (R, <, S'), such that f( 0:) ..... Y E Y. Since Y is locally closed, y has
a neighborhood N in Y that is definable in (R, <. S) and closed and bounded ill
R n . To show 0: is cornpletable in X we may as well assume that f( 0:) lies entirely
in N. Then 0: lies in f-1(N) and f-1(N) is closed and bounded in R m , so 0 is
completable in f-1(N), and hence in X.
3. Suppose f is definably proper. Let S  X be closed in X. To show I(H) is
closed in Y, write S = ni Si where (Si)iE/ is a family of definable closed subseh of
X such that for any two indices i 1 and i 2 there is an index i with SI  Si, n H".
It suffices to show that f(S) = n;!(Si)' Let qEnJ(Si), so that f-l(q)n Si is it
closed nonempty subset of f-1(q) for all i. Since f-1(q) is compact, it follows that
the sets f-1(q) n SI have a common point p, so pES and f(p) = q.
Chapter 7, (4.3)
1. Let q,: Rm+l --+ {x E Rm+1: IIxll < I} be the definable homeomorphism gi\'l'::
by </>(x) = xl ../ 1 + IIx11 2 . Use that U E sm is an asymptotic direction for A if iwd
only if uE cl(r/>(A»). Then apply theorem (1.8) from Chapter 4 to c;6(A).
3. L'se the fact that a basic semilinear set in R m is an open subset of an affill<'
subspace of R m , and that R m cannot be covered by finitely many linear subspa('('s
of dimension < m.
Chapter 8, (2.14)
2. (Bounded semilinear sets are polyhedrons.) Along the same lines as the proof of
the triangulation theorem. Assume the desired result holds for a certain value of
m, and let S, S1,. .., Sk be bounded semilinear sets in Rm, Sj  S. Replacing S hv
its closure (and adding the old S to the list of distinguished subsets) we may as well
assume that S is closed. Let l' := bd(S) U bd(Sl) U" . U bd(Sk), so T is closed and
bounded of dimension < m + 1. After applying a linear automorphism of Rm+1 we
may assume by exercise 3 of Chapter 7, (4.3) that e m +1 is a good direction VE'ctor
for T.

HINTS AND SOLUTIO:-lS
171
By the inductive hypothesis, and by Chapter 1, (7.4), there are a closed complex
K in R m and a finite collection F of affine functions on R m , such that IKI = 1I"(T),
the restrictions to each q E K of the functions in F can be arranged in increasing
order as 1".1 < ... < I".j(,,)' and T = u{r(f",j): (J E K, 1  j  j(q)}. We
also take care that K is compatible with all sets 1I"(S n r(fn and 1I"(S; n r(f))
(f E F, 1  i  k). We now indicate how to use the arguments in the proof of
lemma (2.8) to get a complex L in Rm+1 such that K = {1I"(q): qEL}, ILl = S
and each S; is a union of simplexes of L.
Indeed, let A  Rm be any closed and bounded semilinear set, F any finite
nonempty collection of affine functions on Rm, and K any closed complex in R m
with A = IKI, such that for each (J E K the restrictions of the functions in F to (1
can be arranged in increasing order as 1".1 < ... < I",j(,,)' (It may happen that
different functions in F have the same restriction to q.)
Given two successive functions 1 = I".i and 9 = 1",1+1 we note that for at least
one vertex p of q we must have cI(f)(p) < cI(g)(p), where cI(f), cI(g) denote the
continuous extensions of 1 and 9 to cI(q). Fixing a linear order on Vert(K) we
obtain in this way for each successive pair I, 9 as above a closed complex L(f, g) in
R m +1 such that IL(f,g)1 is the convex hull of
{cl(f)(p) : p a vertex of q} U { cl(g )(p) : p a vertex of q},
which equals [cI(f),cI(g)], and also for each 1 = I",i a closed complex L(f) in
Rm+1 with IL(f)1 = r(cl(f)).
All this is just as in the proof of lemma (2.8). Finally, the union of the L(f, g)'s
and L(f)'s is a closed complex in R m , and this complex is compatible with the sets
in K F := set of the (f,g)'s and r(f)'s.
3. Let R. be an L-structure, where L extends the language of ordered rings. We
may as well assume that S is defined by an L- formula 4>( x, y), x = (Xl,"" X m ),
y = (Yl,..., Yn), since the constants from R that may be involved in an L(R)-
formula defining S can be replaced by extra parametric variables (increasing m).
Let R.' be any L-structure elementarily equivalent to R.. Then R.' is also o-minimal,
by Chapter 3, (3.8), and hence satisfies triangulation. Therefore, given any a E R' m
t here is a definable homeomorphism ha from 4>( a, R' n) onto a union of faces of the
simplex (el>" ., eJV) in R' JV, for some N = N(a) E N. Because Th(R.) has definable
Skolem functions, we may further assume the graph of ha is defined by a formula
Ib(a, Y, z) where 1/J(x, y, z) is an L-formula depending on a and with z = (Zl>"" zJV).
Since R.' F= Th(R.) was arbitrary it follows by model-theoretic compactness that
only finitely many different such formulas 1/J and numbers N(a) EN are needed when
R.' varies over all models of Th(R.) and a over R' m. We can easily construct from
these finitely many Ib's a single definable map 1 as required, with N the maximum
of the finitely many numbers N(a).

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Appl. Alj/;.bra 96 (1994), 173-201.

176
60. H. Sussmann, Real analytic df.'siugularization and subanalytic sets: An elementary approw It,
Trans. AMS 317 (1990),417-461.
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Rand Corporation, Berkeley and Los Angeles, 19.11.
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to their probabi/itie.. Th. Prob. Appl. 16 (1971), 264-280.
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Champaign.

additivE' complexity 151, 152
affine coordinates 120
affine definable spar,e 157
affine function 26
affine independent 119
affine map 126
affine span 119
affine subspace 119
Artin, E. 8, 21,
AsrhenbrennE'r, M. vii
asymptotic direction 118
atom 12
attaching 165
barycE'ntpr 120
harycentric coordinates 120
barycentric subdivision 123, 129
basic function 21
hasic relation 21
basic splllilinear set 26
IIenpdetti, R. vii, 7, 8, 41,141,
150-152,154
BiE'rstone, E. 8
Bochnak J. i'!, 41, 106
hoolean algebra 2, 12
Borel set 1, 16
boundary x, 18
hounded 95
Buurbaki, N. 106
hox 17
Brown, A. k, 41,118
Brumfiel, G. 8, 106, 118, 155, 168
Cantor set 1
Cantor's paradise 1
Cantor space 1
Caratheodory, C. 126
('I-cell 114
('I.cell decomposition 115
('I-map 11 L 114
(,'N-c,elJ 116
CN. map 116
CN-celJ decomposition 116
INDEX
celJ :, 50, 51
cell decomposition theorem 4, 49-57
Chanp;, C. 8, 29
chart 156
Chervonenkis, A. 91
Chevalley's constructibility theorem
la, :J4, 76
dosed complex 121
closed multivalued function 128
closure x, 2, 15, 17
codomain ix
compatible ix, 127
rompletable 102, 159
com pletion 16a
complex 121
cone 149
conjunction 11
constant 21
constant symbol 22
constructible set 1 :3, 76
continuity of roots :J2
convex 17, 120
convex hull 120
Cunway's field of surreal numbers 8
Coste, M. 8, 41, 106, 154
curve selection 93, 94, 97
decomposition 51,52
decomposition above :38
decreasing 16
definable additive map 28
definable atlas 156
definable choice 94
definable collection 5
dpfinable contraction 134, 135
definable curve 102, 1.59
definable equivalence relation 94, 97
definable family 5, 59
definable function 3, 19, 156
definable homeomorphism type 133
definablp homotopy 134
definable map 3, 18, 22, 158
definable partition of unity 102
definabJe path 100
177

178
definable point 22
definable quotient 161-168
definable retraction 1:35
dpfinablE' SE't 3, 18, 22,43, 156
definablE' Set of representatives 94, 97
definable Skolem functions 94, 106
definable space 155-161, 168
definable stronp; deformation
retraction 1:J5
definable suhspacE' 158
definable trivialization 142, 147
definable uing constants 22
definablp using parameters 22
definably connected set 19, 51
definably c,onnected component 57
definably contractible 134
definably equivalent sets 7, 132
definably homeomorphic sets 13:J
definably identifyinl!; map 104
definably proper map 104
definably proper over 166
definably proper quotient 162
definably trivial map 7, 142
definably trivial over 142, 147
Dplfs, H. 7,8, 106, 138, 154, 16
DelzE'll, C. 41, 106
Denef, J. 8, 77
dense 17
dependent 82, 84
dependency index 8:J
derivative 12, 107, 1013
descriptive set theory 16
Dieudonne, .J. 138
clifferentiablp 107, 1013
dimension .'), 6:J-69
disjoint sum 164
disjunction 11
division rinp; 20, 28
domain ix
Dudley, R. 84, 91
Efroymson, G. H, 41, 173
embedded hompomorphism typE' 149
embedded definable homeomorphism
typp 149
equivalence of definable atlases 1.')6
JI'DEX
Esquisse d'un Programme vii, 8,
Euler characteristic 5, 69-76
existential quantification 11
expansion 2:3
extreme point 120
face 120, 128
family ix, 5
fewnomial 150
fibE'r 59
fihprwisE' properties 913
finitE'nes lemma 46-49
Finkler, J. vii
fixed point theorem 97
flag 123
formula 11
Fourier-Motzkin plimination 29
fron tier x, 66
Fuchs, L. 27, 29
full mllhivalued function 128
function ix
function symbol 22
Gabrielov, A. 8
p;enerir triviality 148
Uiesecke, B. 138
I!;luinp; 1.56, 16
I!;ood dirertion 6, 117, Ul, 142
graph ix
GrothE'ndiE'ck, A. vii, H, 9
Hardt, R. 141, 154
Hausdorff space x, 95, 106, 159
Henson, C.W. vii
Hironaka, H. 8, 138
Holly, J. vii
implicit function throrem 113
increasing 16
independent 82, 84
indisrf'rnible
interpretation 22
interval 2, 17

interval topology 17
interior x, 2, 17
invprse function theurem 112
Iovino, .J. vii
Jacobian matrix 108
Kechris, A. 16
Keislpr, H. 8, 29
Khovanskii, A. 9, 77, 141, 168
Knebusch, M. 7,8, 106, 138, 139,
154,168
Knight, J. 61
Kuopman, II. 8,41, 11i'!
Krpispl, G. vii
lanp;uap;e 22, 2:3
Laskowski, C. 85,91
limit x, 15,46
Lpwenberp;, A. vii
L'HapitaJ's rule 114
lifting of a trianp;uJation 129
[inp 119
local dimension 69
lucally bounded 97
locally c10spd 51
Lojasiewicz, S. 8, :n, 33, 34, 41, 138
Lojasiewicz propE'rty 37
L-structure 22
Macintyrp, A. vii, 9, 91
MacPherson, R. 8
map ix
Marker, D. vii, 9
M atyasevic.h's thE'orem 16
Miller, C. vii, i'!, 9,41
Milman, P. 8, In
modpl cOlllplptpncss 13
model-theoretic structurp 21
monotone 16
monotonicity theorem 3, 43-46
morphism of definablp spacE'S 157
multivaluE'd function 128
INDEX
17
negation 11
!\esin, A. vii
!\oethPT normalization 6,39, 138
a-minimal structure 2, 6,16,17,23
opE'n cell 50
ordered division ring 27, 28
ordE'recl field 21
ordered field of real numbers :n
orderE'd F-linear space 25
ordered !!;Toup 19
ordered ring 20
partition ix, 51, 52
Peterzil, Y. 3, 106
Pfaffian functions 9
PilIay, A. vii, 8, 29, 61,77,91, 168
plane 119
Poincare, H. 1
poJyhpdron 121,133,170
polynomially bounded 41
primitive 21
product 158
projective space 1,')7
proper map 106
Pukhlikov A. 77
quantifier elimination l::J, 29
Ramsey's theurem 86
real closed fiE'ld 7, 21, 76
real closure R
refinemen t ix
rep;ular space 158
relation symbol 22
restriction ix, ],5
rin viii, 15
Risler, .I.-J. vii, 7,8,41,141,
150-152, 154
Robson, R. l.j5, 159, 168
Roy, M.-F. R, 41,106

180
Schanue], S. vii, 77
Schreier, O. H, 21
Scowcroft, P. 106
semial!!;ebraic map 40
spmialgebraic spt 1,2,24,31-41
semiaJ!!;ebraic cpll decomposition :J:3,
:J5
spmianalytic set 13
spmilinear spt 1, 2, 25, 26
spmilinpar map 119
sheaf I,'), 156
She]ah, S. 79,85,91
Shiota, M. 9, 154
si!!;n condition :32
simp]p function 24
simple set 24
simp]px 119
Sonta!!;, E. 29,91
Speisseg!!;er, P. vii, 106
star 135
Starchenko, S. ii, 29
Steinhorn, C. vii, 8, 29, 61, 91, 106
Sten!!;le, G. 91
stratification 68
strictly decreasinp; 16
strictly incrpasinp; 16
strictly monotonp 16
structure 1:3
Strzpbonski, A. 77
subanalytie set 2, 8
sub complex 121
Sussmann, H. 1'\, 118, 1:38
SwiPTczkowski, S. vii
Taylor's formuJa 114
tame topology 8, 9
Tarski, A. 8, 21
Tarski-Spidpnbpr!!; theorem 2, 31, :31,
41
Tpissier, II. 9
Thom's lemma :H
Thom's lemma with parametprs :J8
TietzI' pxtension lemma 7, 138
trianp;ulation ] 27
trian!!;ulated set 127
trian!!;uJation throrpm 7, 122, 130
INDEX
trivialization theorpm 7, 147
trivial map 141
underlying set 21
uniform finiteness property 53
uniform continuity 97
universal quantification 12
Van dpn Drips, L. vii, 13, 9, 61, !Of;
Vapnik, V. 91
Vapnik-Chprvonpnkis class 6, HI
Vapnik-Chervonenkis index 81
Vapnik-Chervonenkis property vii.
79 91
VC-class 6,81
verteX map 122
vprtices 120
Viro, O. 77
Whitney, H. 41
Wilkie, A. vii, 2, 9, 91, 150
Wilkie's Theorpm vii, 3, 7, 8,9, :ri.
141, 1.54
Woprheidp, A. vii, 13H
Yukich, .J. 91
Zil'ber Principle 29