Submission date: 4 January 2008.

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 weakly o-minimal structures. 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 of Macpherson, Marker and Steinhorn for sets and functions definable in models of weakly o-minimal theories. In the paper we also investigate large definable subsets of cartesian products of definable sets. Finally, we find various conditions equivalent to the fact that the topological dimension has the addition property.

Mathematics Subject Classification: 03C64

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