Real Algebraic and Analytic Geometry
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181. Roman Wencel:
Topological properties of sets definable in weakly o-minimal structures.

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Submission: 2008, January 4.

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 \cite{MMS} 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 $X\times Y$, then there are many tuples $\langle\overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$, where $k=\mathrm{dim}(Y)$, such that the union of fibres $S_{\overline{a}_{1}}\cup\ldots\cup S_{\overline{a}_{2^{k}}}$ is large in $Y$. Finally, given a weakly o-minimal structure $\mathcal{M}$, we find various conditions equivalent to the fact that the topological dimension in $\mathcal{M}$ enjoys the addition property.

Mathematics Subject Classification (2000): 03C64.

Full text, 23p.: dvi 176k, ps.gz 206k, pdf 273k.