Following their introduction in the early 1980s, o-minimal structures have provided an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. This book gives a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. It starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.
1. Some elementary results; 2. Semialgebraic sets; 3. Cell decomposition; 4. Definable invariants: Dimension and Euler characteristic; 5. The Vapnik–Chernovenkis property in o-minimal structures; 6. Point-set topology in o-minimal structures; 7. Smoothness; 8. Triangulation; 9. Trivialization; 10. Definable spaces and quotients.
"...indispensable to any student or research interested in o-minimal structures. ...written with remarkable precision and clarity..." Bulletin of the AMS
"This is an excellent textbook..." Bulletin of Symbolic Logic