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.

Keywords

• galaxy
• halo
• metric completion
• nonstandard hull
• universal cover
• heine-borel property
• 53a99
• 26e35
• 03h05