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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.