A trichotomy theorem for o‐minimal structures

Let M = 〈 M , <, …〉 be alinearly ordered structure. We define M to be o‐minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo‐minimal M can induce on a neighbourhood of any a in M . Roughly said, one of the following holds: (i) a is trivial (technical term), or(ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or(iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification : primary 03C45; secondary 03C52, 12J15, 14P10.