Compactness in Metric Spaces

• Autorzy: Kazuhisa Nakasho , Keiko Narita , Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

metric spaces; normed linear spaces; compactness

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2016
• Tom: 24
• Numer: 3
• Strony: 167-172

Daty

wydano

2016-09-01

otrzymano

2016-06-30

online

2017-02-08

Twórcy

autor

Kazuhisa Nakasho

• Shinshu University Nagano,

autor

Keiko Narita

• Hirosaki-city Aomori,

autor

Yasunari Shidama

• Shinshu University Nagano,

Identyfikatory

DOI

10.1515/forma-2016-0013