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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| Published in: | Formalized mathematics Vol. 24; no. 3; pp. 167 - 172 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Bialystok
Sciendo
01.09.2016
De Gruyter Brill Sp. z o.o., Paradigm Publishing Services |
| Subjects: | |
| ISSN: | 1898-9934, 1426-2630, 1898-9934 |
| Online Access: | Get full text |
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| 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]. |
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| Bibliography: | 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 |