My aim is to show that o-minimal structures provide an excellent framework for developing tame topology, or topologie modérée, 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 o-minimal the subject has grown rapidly. The supply of ominimal 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 Illinois 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.