an:06255214
Zbl 1309.14046
Kocel-Cynk, Beata; Paw??ucki, Wies??aw; Valette, Anna
A short geometric proof that Hausdorff limits are definable in any o-minimal structure
EN
Adv. Geom. 14, No. 1, 49-58 (2014).
00329029
2014
j
14P10 32B20 03C64 14P15
Hausdorff limit; o-minimal structure
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 \textit{D. Marker} and \textit{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. \textit{J. M. Lion} and \textit{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)]).
Gareth Jones (Manchester)
Zbl 0801.03026; Zbl 1059.03031; Zbl 1222.32019