Real Algebraic and Analytic Geometry |

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.:
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ps.gz 206k,
pdf 273k.

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