Real Algebraic and Analytic Geometry
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Submission: 2012, October 10.
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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