Journal of Symbolic Logic

Topological properties of sets definable in weakly o-minimal structures

Roman Wencel

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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.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 3 (2010), 841-867.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682203

Digital Object Identifier
doi:10.2178/jsl/1278682203

Mathematical Reviews number (MathSciNet)
MR2723770

Zentralblatt MATH identifier
1200.03023

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality

Citation

Wencel, Roman. Topological properties of sets definable in weakly o-minimal structures. J. Symbolic Logic 75 (2010), no. 3, 841--867. doi:10.2178/jsl/1278682203. https://projecteuclid.org/euclid.jsl/1278682203


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