Kocel-Cynk, Beata; Pawłucki, Wiesław; Valette, Anna A short geometric proof that Hausdorff limits are definable in any o-minimal structure. (English) Zbl 1309.14046 Adv. Geom. 14, No. 1, 49-58 (2014). 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)]). Reviewer: Gareth Jones (Manchester) Cited in 2 Documents 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 Keywords:Hausdorff limit; o-minimal structure Citations:Zbl 0801.03026; Zbl 1059.03031; Zbl 1222.32019 PDFBibTeX XMLCite \textit{B. Kocel-Cynk} et al., Adv. Geom. 14, No. 1, 49--58 (2014; Zbl 1309.14046) Full Text: DOI Link