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Constructing metrics with the Heine-Borel property

Authors: Robert Williamson and Ludvik Janos
Journal: Proc. Amer. Math. Soc. 100 (1987), 567-573
MSC: Primary 54E35; Secondary 54E50
MathSciNet review: 891165
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Abstract: A metric space $ (X,d)$ is said to be Heine-Borel if any closed and bounded subset of it is compact. We show that any locally compact and $ \sigma $-compact metric space can be made Heine-Borel by a suitable remetrization. Furthermore we prove that if the original metric $ d$ is complete, then this can be done so that the new Heine-Borel metric $ d'$ is locally identical to $ d$, i.e., for every $ x \in X$ there exists a neighborhood of $ x$ on which the two metrics coincide.

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Keywords: Heine-Borel property, uniformly locally compact metric space, paracompact space
Article copyright: © Copyright 1987 American Mathematical Society

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