On Some Aspects of Compactness in Metric Spaces

Abstract

In this paper, we investigate the generalizations of the concepts from Heine-Borel Theorem and the Bolzano-Weierstrass Theorem to metric spaces. We show that the metric space X is compact if every open covering has a finite subcovering. This abstracts the Heine-Borel property. Indeed, the Heine-Borel Theorem states that closed bounded subsets of the real line R are compact. In this study, we rephrase compactness in terms of closed bounded subsets of the real line R, that is, the Bolzano-Weierstrass theorem. Let X be any closed bounded subset of the real line. Then any sequence (xn) of the points of X has a subsequence converging to a point of X. We have used these interesting theorems to characterize compactness in metric spaces.

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APA

Amonyela</div>, < (2024). On Some Aspects of Compactness in Metric Spaces. Afribary. Retrieved from https://afribary.com/works/on-some-aspects-of-compactness-in-metric-spaces

MLA 8th

Amonyela</div>, <div>Isabu "On Some Aspects of Compactness in Metric Spaces" Afribary. Afribary, 04 Jun. 2024, https://afribary.com/works/on-some-aspects-of-compactness-in-metric-spaces. Accessed 07 Sep. 2024.

MLA7

Amonyela</div>, <div>Isabu . "On Some Aspects of Compactness in Metric Spaces". Afribary, Afribary, 04 Jun. 2024. Web. 07 Sep. 2024. < https://afribary.com/works/on-some-aspects-of-compactness-in-metric-spaces >.

Chicago

Amonyela</div>, <div>Isabu . "On Some Aspects of Compactness in Metric Spaces" Afribary (2024). Accessed September 07, 2024. https://afribary.com/works/on-some-aspects-of-compactness-in-metric-spaces

On Some Aspects of Compactness in Metric Spaces

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