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Constructing Metrics with the Heine-Borel Property

Robert Williamson and Ludvik Janos
Proceedings of the American Mathematical Society
Vol. 100, No. 3 (Jul., 1987), pp. 567-573
DOI: 10.2307/2046449
Stable URL: http://www.jstor.org/stable/2046449
Page Count: 7
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Constructing Metrics with the Heine-Borel Property
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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 σ-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 ∈ X there exists a neighborhood of x on which the two metrics coincide.

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