Definable Compactness and Definable Subgroups of o-Minimal Groups |
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Authors: | Peterzil Ya'acov; Steinhorn Charles |
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Institution: | Department of Mathematics and Computer Sciences, University of Haifa Haifa, Israel
Department of Mathematics, Vassar College Poughkeepsie, NY 12601, USA |
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Abstract: | The paper introduces the notion of definable compactness andwithin the context of o-minimal structures proves several topologicalproperties of definably compact spaces. In particular a definableset in an o-minimal structure is definably compact (with respectto the subspace topology) if and only if it is closed and bounded.Definable compactness is then applied to the study of groupsand rings in o-minimal structures. The main result proved isthat any infinite definable group in an o-minimal structurethat is not definably compact contains a definable torsion-freesubgroup of dimension 1. With this theorem, a complete characterizationis given of all rings without zero divisors that are definablein o-minimal structures. The paper concludes with several examplesillustrating some limitations on extending the theorem. |
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