## Abstract

Let script M sign = 〈M, <, . . .〈 be a linearly ordered structure. We define script M sign 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 arbitrary o-minimal script M sign 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 script M sign induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which script M sign 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.

Original language | English |
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Pages (from-to) | 481-523 |

Number of pages | 43 |

Journal | Proceedings of the London Mathematical Society |

Volume | 77 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1998 |

### Bibliographical note

Funding Information:The work of the first author was partially supported by an SERC grant. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

## ASJC Scopus subject areas

- Mathematics (all)