O-minimal theory

Summary

An o-minimal structure is a totally ordered structure in which every definable subset of the line is a finite union of points and open intervals. An o-minimal theory is a theory all of whose models are o-minimal. This condition on sets in one variable forces tame behaviour in all dimensions. Definable sets admit cell decompositions and have a good dimension theory. The ordered field of real numbers is the model case. O-minimality is the basis of "tame geometry", used in real algebraic geometry and in number theory.