Compactness

Surinder Pal Singh Kainth ()
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Surinder Pal Singh Kainth: Panjab University, Department of Mathematics

Chapter Chapter 5 in A Comprehensive Textbook on Metric Spaces, 2023, pp 123-154 from Springer

Abstract: 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.

Date: 2023
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DOI: 10.1007/978-981-99-2738-8_5

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