FUZZY METRIC SPACES - KAIST.pdfVIP

J. Appl. Math. Computing Vol. 16(2004), No. 1 - 2, pp. 371 - 381 FUZZY METRIC SPACES ZUN-QUAN XIA AND FANG-FANG GUO∗ Abstract. In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition. AMS Mathematics Subject Classification: 03E72, 90C70, 15A03. Key words and phrases : Fuzzy metric space, completeness of fuzzy metric space, fuzzy topology, fuzzy closed set. 1. Introduction How to define fuzzy metric is one of the fundamental problems in fuzzy mathe- matics which is wildly used in fuzzy optimization and pattern recognition. There are two approaches in this field till now. One is using fuzzy numbers to define metric in ordinary spaces, firstly proposed by Kaleva (1984)[12], following which fuzzy normed spaces, fuzzy topology induced by fuzzy metric spaces, fixed point theorem and other properties of fuzzy metric spaces are studied by a few re- searchers, see for instance, Felbin (1992)[7], George (1994)[8], George (1997)[9], Gregori (2000)[10], Hadzic (2002)[11] etc. The other one is using real numbers to measure the distances between fuzzy sets. The references of this approach can be referred to, for instance, Dia (1990) [5],