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    On Some Aspects of Compactness in Metric Spaces

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    On Some Aspects of Compactness in Metric Spaces.pdf (268.5Kb)
    Date
    2024-04-17
    Author
    Isabu, Hillary Amonyela
    Ojiema, Michael Onyango
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    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.
    URI
    https://doi.org/10.51867/Asarev.Maths.1.1.2
    https://asarev.net/ojs/index.php/asarev/article/view/2
    http://ir-library.mmust.ac.ke:8080/xmlui/handle/123456789/2853
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