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Suppose (X,d) is a bounded metric space. Can we extend (X,d) into (X',d') such that (X',d') is compact and d and d' agree on X?

( The reason for asking the question: To prove a theorem in Euclidean space, I found it convenient to first extend the bounded set in question to a compact one ( its closure, to be exact.) and work with the latter. The proof can be generalized to complete spaces, but to go any further I need the answer to this question.)

Thanks in advance.

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# Extending Bounded metric spaces to compact spaces

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