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Topological properties of sets definable in weakly o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Roman Wencel
Roman Wencel*
Affiliation:
Mathematical Institute, University of WrocławPl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. E-mail: [email protected]
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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 , where k = dim(Y), the union of fibers is large in Y. Finally, given a weakly o-minimal structure , we find various conditions equivalent to the fact that the topological dimension in enjoys the addition property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 841 - 867
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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