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On completion of fuzzy metric spaces. (English) Zbl 1010.54002
Summary: Completions of fuzzy metric spaces (in the sense of A. George and V. P. Veeramani [ibid. 64, No. 3, 395-399 (1994; Zbl 0843.54014); 90, No. 3, 365-368 (1977; Zbl 0917.54010)]) are discussed. A complete fuzzy metric space \(Y\) is said to be a fuzzy metric completion of a given fuzzy metric space \(X\) if \(X\) is isometric to a dense subspace of \(Y\). We present an example of a fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a dense subspace uniformly isomorphic to it.

MSC:
54A40 Fuzzy topology
54E35 Metric spaces, metrizability
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