An introduction to o-minimal structures

TL;DR

This paper introduces o-minimal structures, highlighting their development since the 1980s and their broad applications in geometry, with examples and properties of definable sets and maps.

Contribution

It provides an accessible overview of o-minimal structures, emphasizing their geometric aspects and examples without delving into the model-theoretic details.

Findings

O-minimal structures generalize semialgebraic and subanalytic geometry.

Definable sets in o-minimal structures have well-behaved geometric properties.

The paper presents several examples illustrating the scope of o-minimal structures.

Abstract

The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal structures. We omit here any reference to the pure model theory of o-minimal structures and to the theory of groups and rings definable in o-minimal structures.