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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 24, Issue 3 (Sep 2016)

Issues

Compactness in Metric Spaces

Kazuhisa Nakasho / Keiko Narita / Yasunari Shidama
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0013

Summary

In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces.

In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].

MSC: 46B50; 54E45; 03B35

Keywords: metric spaces; normed linear spaces; compactness

MML: identifier: TOPMETR4; version: 8.1.05 5.37.1275

References

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About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0013.

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© by Kazuhisa Nakasho. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License BY-SA 3.0 LEGALCODE

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