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Completeness

In: A Comprehensive Textbook on Metric Spaces

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  • Surinder Pal Singh Kainth

    (Panjab University, Department of Mathematics)

Abstract

Roughly speaking, a metric space X is complete if every sequence in X, which attempts to converge, finds a buddy in X for that purpose. In other words, X is incomplete if it lacks some ‘good’ points. However, it is always possible to extend an incomplete space to a complete one, by appending all such missing ‘good’ points. This chapter starts with a brief introduction to complete metric spaces, followed by its most important application; the Banach Contraction Principle. Then we provide various characterizations of completeness, in terms of Cantor intersection property and totally bounded sets. The completion of a metric space is discussed in a separate section where we establish that the Cauchy completion of $$\mathbb {Q}$$ is isometric to its Dedekind completion. Finally, we present various Banach spaces, including the space of continuous functions, and some results regarding absolute and unconditional convergence.

Suggested Citation

  • Surinder Pal Singh Kainth, 2023. "Completeness," Springer Books, in: A Comprehensive Textbook on Metric Spaces, chapter 0, pages 89-121, Springer.
    • Handle: RePEc:spr:sprchp:978-981-99-2738-8_4
    DOI: 10.1007/978-981-99-2738-8_4
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