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Compact and Locally Compact Spaces

In: Measure-Theoretic Calculus in Abstract Spaces

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  • Zigang Pan

Abstract

The concept of compactness of a set is pervasive in analysis. This chapter is devoted to the study of compact spaces and locally compact spaces. The concept of compactness involves actually four concepts: countably compact, Bolzano–Weierstrass property, sequentially compact, and compact for general topological spaces. The metric space again demonstrates its superiority over topological spaces, where all these compactness notions are equivalent in metric spaces as demonstrated in Borel–Lebesgue Theorem 5.37. We present the following main results on compact spaces: Ascoli–Arzelá Theorem 5.44, Tychonoff Theorem 5.47, partition of unity (Theorem 5.63), Alexandroff one-point compactification theorem 5.66, and Stone–Čech compactification (Proposition 5.83).

Suggested Citation

  • Zigang Pan, 2023. "Compact and Locally Compact Spaces," Springer Books, in: Measure-Theoretic Calculus in Abstract Spaces, chapter 0, pages 105-136, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-21912-2_5
    DOI: 10.1007/978-3-031-21912-2_5
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