Compactness in Metric Spaces

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 space...

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Veröffentlicht in:Formalized mathematics Jg. 24; H. 3; S. 167 - 172
Hauptverfasser: Nakasho, Kazuhisa, Narita, Keiko, Shidama, Yasunari
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Bialystok Sciendo 01.09.2016
De Gruyter Brill Sp. z o.o., Paradigm Publishing Services
Schlagworte:
03B35
46B50
54E45
compactness
identifier: TOPMETR4
metric spaces
normed linear spaces
version: 8.1.05 5.37.1275
ISSN:1898-9934, 1426-2630, 1898-9934
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Zusammenfassung: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].
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1898-9934
1426-2630
1898-9934
DOI:10.1515/forma-2016-0013