Metric completions, the Heine-Borel property, and approachability

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.

8 pages, to appear in Open Mathematics

References in corpus (2)

• Cauchy, infinitesimals and ghosts of departed quantifiers
• Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context