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Some properties of fuzzy metric spaces. (English) Zbl 0985.54007
One of the main problems in the theory of fuzzy topological spaces is to obtain an appropriate and consistent notion of fuzzy metric space. Many authors have investigated this question and several different notions of a fuzzy metric space have been defined and studied. In [{\it A. George} and the reviewer, ibid. 64, No. 3, 395-399 (1994; Zbl 0843.54014)] by modifying the concept of metric fuzziness introduced by {\it I. Kramosil} and {\it J. Michalek} [Kybernetika, Praha 11, 336-344 (1975; Zbl 0319.54002)] an interesting notion of fuzzy metric space has been studied. In [J. Fuzzy Math. 3, No. 4, 933-940 (1995; Zbl 0870.54007)] {\it A. George} and the reviewer, have proved that every such metric space is metrizable. In this paper the authors give a more natural proof for this result. In addition to this result the authors prove some interesting results like: every separable fuzzy metric space admits a compatible precompact fuzzy metric and, a fuzzy metric space is compact if ond only if it is precompact and totally bounded. In particular, the celebrated Niemytzki-Tychonoff theorem is generalized to the fuzzy setting.

54A40Fuzzy topology
54E35Metric spaces, metrizability
Full Text: DOI
[1] R. Engelking, General Topology, PWN-Polish Sci. Publ., Warsaw, 1977.
[2] George, A.; Veeramani, P. V.: On some results in fuzzy metric spaces. Fuzzy sets and systems 64, 395-399 (1994) · Zbl 0843.54014
[3] George, A.; Veeramani, P. V.: On some results of analysis for fuzzy metric spaces. Fuzzy sets and systems 90, 365-368 (1997) · Zbl 0917.54010
[4] Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy sets and systems 27, 385-389 (1989) · Zbl 0664.54032
[5] J.L. Kelley, General Topology, Van Nostrand, 1955.
[6] Kramosil, O.; Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetica 11, 326-334 (1975) · Zbl 0319.54002
[7] Schweizer, B.; Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 314-334 (1960) · Zbl 0091.29801