Constructing metrics with the Heine-Borel property

Williamson, Robert; Janos, Ludvik

doi:10.1090/S0002-9939-1987-0891165-X

Constructing metrics with the Heine-Borel property
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by Robert Williamson and Ludvik Janos PDF

Proc. Amer. Math. Soc. 100 (1987), 567-573 Request permission

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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