Metric completions, the Heine-Borel property, and approachability-Reference-Cited by-同舟云学术

Metric completions, the Heine-Borel property, and approachability

Published:2020-01-01 Issue:1 Volume:18 Page:162-166
ISSN:2391-5455
Container-title:Open Mathematics
language:en
Short-container-title:

Author:

Kanovei Vladimir¹,Katz Mikhail G.²,Nowik Tahl²

Affiliation:

1. IPPI RAS , Moscow , Russia

2. Department of Mathematics , Bar Ilan University , Ramat Gan , 5290002 , Israel

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/math-2020-0017/pdf

Reference20 articles.

1. Robert Goldblatt, Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Texts in Mathematics 188, Springer-Verlag, New York, 1998.

2. Martin Davis, Applied nonstandard analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Reprinted by Dover, NY, 2005. Available: http://store.doverpublications.com/0486442292.html

3. Abraham Robinson, Non-standard analysis, North-Holland Publishing, Amsterdam, 1966.

4. Tahl Nowik and Mikhail Katz, Differential geometry via infinitesimal displacements, J. Log. Anal. 7 (2015), no. 5, 1–44. Available: http://www.logicandanalysis.org/index.php/jla/article/view/237/106 and http://arxiv.org/abs/1405.0984

5. Vladimir Kanovei, Karin Katz, Mikhail Katz, and Tahl Nowik, Small oscillations of the pendulum, Euler’s method, and adequality, Quantum Stud. Math. Found. 3 (2016), no. 3, 231–236. Available: http://dx.doi.org/10.1007/s40509-016-0074-x and https://arxiv.org/abs/1604.06663