On Some Aspects of Compactness in Metric Spaces

Hillary Amonyela  Isabu; Michael Onyango Ojiema

doi:10.51867/Asarev.Maths.1.1.2

Authors

Hillary Amonyela Isabu Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya. Author https://orcid.org/0000-0001-9305-8233
Michael Onyango Ojiema Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya Author https://orcid.org/0000-0001-9635-7597

DOI:

https://doi.org/10.51867/Asarev.Maths.1.1.2

Keywords:

Metric space, Bolzano-Weierstrass theorem, Heine-Borel theorem

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 (x_n) 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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