## Abstract

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Original language | English |
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Pages (from-to) | 769-786 |

Number of pages | 18 |

Journal | Journal of the London Mathematical Society |

Volume | 59 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1999 |

### Bibliographical note

Funding Information:The first author was partially supported by an SERC grant. The second author was partially supported by a visiting research fellowship at Merton College, University of Oxford, and NSF grant DMS-9401723.

## ASJC Scopus subject areas

- Mathematics (all)