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DEFINABLE COMPACTNESS AND DEFINABLE SUBGROUPS OF o-MINIMAL GROUPS

Published online by Cambridge University Press:  01 June 1999

YA'ACOV PETERZIL  and
CHARLES STEINHORN
YA'ACOV PETERZIL
Affiliation:
Department of Mathematics and Computer Sciences, University of Haifa, Haifa, Israel
CHARLES STEINHORN
Affiliation:
Department of Mathematics, Vassar College, Poughkeepsie, NY 12601, USA
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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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 769 - 786
Copyright
The London Mathematical Society 1999

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