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arXiv · 2002.07536

Metric completions, the Heine-Borel property, and approachability

Vladimir Kanovei·Mikhail G. Katz·Tahl Nowik
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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.

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Vladimir Kanovei, Mikhail G. Katz, Tahl Nowik. 2020-02-18. Metric completions, the Heine-Borel property, and approachability. https://doi.org/10.1515/math-2020-0017

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