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Write the Heine – Borel property |
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Answer» 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-ocompact metric space can be made Heine-Borel by a suitable remetrization. Furthermor,e 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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