Definable compactness and definable subgroups of o-minimal groups

Ya'Acov Peterzil, Charles Steinhorn

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)769-786
Number of pages18
JournalJournal of the London Mathematical Society
Issue number3
StatePublished - 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

  • General Mathematics


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