Author(s): Lou van Dries
Series: London Mathematical Society Lecture Note Series. 248
Publisher: Cambridge University Press
Year: 1998
Language: English
Pages: 189
Cover......Page 1
Title page......Page 2
PREFACE......Page 6
PREREQUISITIES......Page 7
CONVENTIONS AND NOTATIONS......Page 8
INTRODUCTION AND OVERVIEW......Page 10
1. Remarks on logical notation and boolean algebras......Page 20
2. Elementary facts on structures......Page 22
3. O-minimal structures......Page 25
4. O-minimal ordered groups and rings......Page 28
5. Model-theoretic structures......Page 30
6. The simplest o-minimal structures......Page 33
7. Semilinear sets......Page 34
Notes and comments......Page 38
1. Thom's lemma and continuity of roots......Page 40
2. Semialgebraic cell decomposition......Page 42
3. Thom's lemma with parameters......Page 47
Notes and comments......Page 50
1. The monotonicity theorem and the finiteness lemma......Page 52
2. The cell decomposition theorem......Page 58
3. Definable families......Page 68
Notes and comments......Page 70
1. Dimension......Page 72
2. Euler characteristic......Page 78
Notes and comments......Page 86
1. A combinatorial dichotomy......Page 88
2. Vapnik-Chervonenkis classes and dependence......Page 90
3. Reduction to the case q == 1......Page 94
Notes and comments......Page 100
1. Curve selection......Page 102
2. Fiberwise properties......Page 107
3. Paths and partitions of unity......Page 109
4. Curves, proper maps, and identifying maps......Page 111
Notes and comments......Page 115
1. Differentiability in ordered fields......Page 116
2. Inverse function theorem......Page 118
3. Definable maps are piecewise C¹......Page 123
4. Existence of good directions......Page 126
Notes and comments......Page 127
1. Simplexes and complexes......Page 128
2. Triangulation theorem......Page 136
3. Definable retractions and definable continuous extensions......Page 143
Notes and comments......Page 147
Chapter 9 TRIVIALIZATION......Page 150
1. Trivialization theorem......Page 151
2. Applications......Page 158
3. On a conjecture of Benedetti and Risler......Page 159
Notes and comments......Page 163
Chapter 10 DEFINABLE SPACES AND QUOTIENTS......Page 164
1. Definable spaces......Page 165
2. Definable quotient spaces......Page 170
Notes and comments......Page 177
HINTS AND SOLUTIONS......Page 178
REFERENCES......Page 182
lNDEX......Page 186
