Metric completions, the Heine-Borel property, and approachability

Vladimir Kanovei; Mikhail G. Katz; Tahl Nowik

doi:10.1515/math-2020-0017

Metric completions, the Heine-Borel property, and approachability

Vladimir Kanovei, Mikhail G. Katz, Tahl Nowik

Department of Mathematics

Russian Academy of Sciences

Research output: Contribution to journal › Article › peer-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 language	English
Pages (from-to)	162-166
Number of pages	5
Journal	Open Mathematics
Volume	18
Issue number	1
DOIs	https://doi.org/10.1515/math-2020-0017
State	Published - 1 Jan 2020

Bibliographical note

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

Funding

We are grateful to the referees for helpful comments on an earlier version of the manuscript. V. Kanovei was partially supported by RFBR grant no 18-29-13037.

Funders	Funder number
Russian Foundation for Basic Research	18-29-13037

Keywords

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

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10.1515/math-2020-0017

Cite this

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Metric completions, the Heine-Borel property, and approachability

Abstract

Bibliographical note

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Keywords

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