Abstract
A metric space ( X , d ) (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 \sigma -compact metric space can be made Heine-Borel by a suitable remetrization. Furthermore we prove that if the original metric d d is complete, then this can be done so that the new Heine-Borel metric d ′ d’ is locally identical to d d , i.e., for every x ∈ X x \in X there exists a neighborhood of x x on which the two metrics coincide.
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