1998/11/01 by Ya’acov Peterzil, Sergei Starchenko · 4 citations
Mathematics · #Advanced Topology and Set Theory #Mathematical and Theoretical Analysis #Rings, Modules, and Algebras
paper · doi:10.1112/s0024611598000549
openalex publication_date 1998/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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.