Hi 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.