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Constructing metrics with the Heine-Borel property. (English) Zbl 0626.54035
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.

MSC:
54E50 Complete metric spaces
54E35 Metric spaces, metrizability
54D45 Local compactness, \(\sigma\)-compactness
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