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September 2010 Topological properties of sets definable in weakly o-minimal structures
Roman Wencel
J. Symbolic Logic 75(3): 841-867 (September 2010). DOI: 10.2178/jsl/1278682203

Abstract

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X× Y, then for a large set of tuples 〈\overline{a}₁,…,\overline{a}2k〉 ∈ X{2k}, where k=dim(Y), the union of fibers S\overline{a}₁∪…∪ S\overline{a}2k is large in Y. Finally, given a weakly o-minimal structure ℳ, we find various conditions equivalent to the fact that the topological dimension in ℳ enjoys the addition property.

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Roman Wencel. "Topological properties of sets definable in weakly o-minimal structures." J. Symbolic Logic 75 (3) 841 - 867, September 2010. https://doi.org/10.2178/jsl/1278682203

Information

Published: September 2010
First available in Project Euclid: 9 July 2010

zbMATH: 1200.03023
MathSciNet: MR2723770
Digital Object Identifier: 10.2178/jsl/1278682203

Subjects:
Primary: 03C64

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 3 • September 2010
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