Metric completions, the Heine-Borel property, and approachability

Vladimir Kanovei, Mikhail G. Katz, Tahl Nowik

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)162-166
Number of pages5
JournalOpen Mathematics
Volume18
Issue number1
DOIs
StatePublished - 1 Jan 2020

Bibliographical note

Publisher Copyright:
© 2020 Vladimir Kanovei et al., published by De Gruyter 2020.

Keywords

  • Heine-Borel property
  • galaxy
  • halo
  • metric completion
  • nonstandard hull
  • universal cover

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