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Compactness

In: A Comprehensive Textbook on Metric Spaces

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  • Surinder Pal Singh Kainth

    (Panjab University, Department of Mathematics)

Abstract

In this chapter, we discuss compactness, its characterizations, and its relationship with continuity. We start with a basic introduction to compact sets, their relationship with closed subsets, and the case of Euclidean spaces. Then we present various characterizations of compactness; in terms of finite intersection property, sequential compactness, and totally bounded sets. We provide a detailed discussion on continuity and compactness; particularly on the uniform and Lipschitz continuities. We also present some necessary and sufficient conditions for the uniform continuity of real functions. Some recent results regarding UC spaces, strong uniform continuity and Cauchy continuous maps, which haven’t yet made their way into any textbook, are also outlined in the exercises.

Suggested Citation

  • Surinder Pal Singh Kainth, 2023. "Compactness," Springer Books, in: A Comprehensive Textbook on Metric Spaces, chapter 0, pages 123-154, Springer.
    • Handle: RePEc:spr:sprchp:978-981-99-2738-8_5
    DOI: 10.1007/978-981-99-2738-8_5
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