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A short geometric proof that Hausdorff limits are definable in any o-minimal structure. (English) Zbl 1309.14046
Suppose that \(\mathcal{A}\) is an o-minimal expansion of the real field. If \(\mathcal{A}\) is an \(\mathcal{R}\)-definable family of nonempty compact subsets of \(\mathbb{R}^n\) then the closure of \(\mathcal{A}\) in the Hausdorff metric is also a definable family. This theorem has several proofs. The original proof by D. Marker and C. I. Steinhorn [J. Symb. Log. 59, No. 1, 185–198 (1994; Zbl 0801.03026)] used model theory. There have also been several other model-theoretic proofs. J. M. Lion and P. Speissegger [Sel. Math., New Ser. 10, No. 3, 377–390 (2004; Zbl 1059.03031)] gave a geometric proof using blowings-up in jet spaces. Here the authors give a new geometric proof, based on the Lipschitz cell decompositions introduced by the second and first authors (see for instance [Ill. J. Math. 52, No. 3, 1045–1063 (2008; Zbl 1222.32019)]).

MSC:
14P10 Semialgebraic sets and related spaces
32B20 Semi-analytic sets, subanalytic sets, and generalizations
03C64 Model theory of ordered structures; o-minimality
14P15 Real-analytic and semi-analytic sets
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