Metric completions, the Heine-Borel property, and approachability
Differential Geometry
2020-03-05 v2 Classical Analysis and ODEs
Logic
Abstract
We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in *M.
Keywords
Cite
@article{arxiv.2002.07536,
title = {Metric completions, the Heine-Borel property, and approachability},
author = {Vladimir Kanovei and Mikhail G. Katz and Tahl Nowik},
journal= {arXiv preprint arXiv:2002.07536},
year = {2020}
}
Comments
8 pages, to appear in Open Mathematics