Real Algebraic and Analytic Geometry |

A short geometric proof that Hausdorff limits are definable in any o-minimal structure.

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Submission: 2012, October 10.

*Abstract:
The aim of this note is to give a short geometric proof of the following
theorem of L. Bröcker [B]: given an arbitrary o-minimal structure on the ordered field
of real numbers R and any definable family A of definable nonempty compact subsets
of Rn, then the closure of A in the sense of the Hausdorff metric (or, equivalently,
in the Vietoris topology) is a definable family. In particular, any limit in the sense
of the Hausdorff metric of a convergent sequence of subsets of a definable family is
definable in the same o-minimal structure. The original proof was based on model
theory. Lion and Speissegger [LS] gave a geometric proof of the theorem. Our proof
here is based on the idea of Lipschitz cell decompositions.*

Mathematics Subject Classification (2010): 14P10, 32B20, 03C64, 14P15.

Keywords and Phrases: Hausdorff limit, o-minimal structure.

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