Metric completions, the Heine-Borel property, and approachability

Abstract

<jats:title>Abstract</jats:title> <jats:p>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 <jats:italic>M</jats:italic> in terms of the absence of inapproachable finite points in <jats:sup>∗</jats:sup><jats:italic>M</jats:italic>.</jats:p>