Preface These notes have served as a basis for a course in Pisa in Spring 1999. A parallel course on the construction of o-minimal structures was given by A. Macintyre. The content of these notes owes a great deal to the excellent book by L. van den Dries [vD]. Some interesting topics contained in this book are not included here, such as the Vapnik-Chervonenkis property. Part of the material which does not come from [vD] is taken from the paper [Co1]. This includes the sections on the choice of good coordinates and the triangulation of functions in Chapter 4 and Chapter 5. The latter chapter contains the results on triviality in families of sets or functions which were the main aim of this course. The last chapter on smoothness was intended to establish property " DC k-all k " which played a crucial role in the course of Macintyre (it can be easily deduced from the results in [vDMi]). It is also the occasion to give a few results on tubular neighborhoods. I am pleased to thank Francesca Acquistapace, Fabrizio Broglia and all colleagues of the Dipartimento di Matematica for the invitation to give this course in Pisa and their friendly hospitality. Also many thanks to Antonio Ponchio for reading these notes and correcting mistakes.
[1]
John L. Bell,et al.
Models and Ultraproducts: An Introduction.
,
1969
.
[2]
Takuo Fukuda.
Types topologiques des polynomes
,
1976
.
[3]
A. Pillay,et al.
Definable sets in ordered structures
,
1984
.
[4]
A. Pillay,et al.
DEFINABLE SETS IN ORDERED STRUCTURES. I
,
1986
.
[5]
Marie-Françoise Roy,et al.
Real algebraic geometry
,
1992
.
[6]
A. Wilkie.
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
,
1996
.
[7]
L. Dries,et al.
Geometric categories and o-minimal structures
,
1996
.
[8]
L. van den Dries,et al.
Tame Topology and O-minimal Structures
,
1998
.
[9]
Michel Coste.
Topological types of fewnomials
,
1998
.
[10]
M. Coste.
AN INTRODUCTION TO SEMIALGEBRAIC GEOMETRY
,
2002
.