Metric Spaces

Zigang Pan

Chapter Chapter 4 in Measure-Theoretic Calculus in Abstract Spaces, 2023, pp 79-103 from Springer

Abstract: Abstract Metric space is a set X together with a nonnegative real-valued function of X × X that defines the notion of “distance” between each pair of points in X. This distance function automatically induces a natural topology on the set X. So, we have all the tools for a topological space at our disposal. The metric space further allows us to talk about uniform continuous, uniform convergence, etc, in general uniformity. Here, the concept of completeness is introduced, which is equivalent to that all Cauchy sequence (net) converges in the space. Thus, a complete metric space doesn’t have “holes” in it. The Baire Category Theorem holds for complete metric spaces. Because of the additional properties of a metric space. We investigate the condition when a topological space can be metricized, which leads to the Urysohn Metrization Theorem. We also present conditions under which two limit operations can be interchanged.

Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-21912-2_4

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DOI: 10.1007/978-3-031-21912-2_4

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