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

Abstract

Abstract. The aim of this note is to give yet another proof of the following theorem: 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 proofs by Bröcker [1], Marker and Steinhorn [7], Pillay [11] (see also van den Dries [15]) were based on model theory. Lion and Speissegger [6] gave a geometric proof of the theorem. Our proof below is based on the idea of Lipschitz cell decompositions.