On fuzzy metric spaces.

*(English)*Zbl 0558.54003This paper proposes a definition of a fuzzy metric space in which the distance between two points is a non-negative, upper semicontinuous, normal, convex fuzzy number. Here fuzzy numbers are as defined by D. Dubois and H. Prade [ibid. 2, 327-348 (1979; Zbl 0412.03035)]. The definition includes a version of the triangle inequality analogous to that of a Menger space [cf. B. Schweizer and A. Sklar, Pac. J. Math. 10, 313-334 (1960; Zbl 0091.298)] but with two bounding functions in place of the t-norm. When Max and Min are the bounding functions, the triangle inequality reduces to the classical form with addition and a partial ordering as defined by M. Mizumoto and K. Tanaka [in ”Advances in Fuzzy Set Theory and Applications”, M. M. Gupta, R. K. Ragade, and R. R. Yager, Eds., North-Holland, New York, 153- 164 (1979; Zbl 0434.94026)]. The authors show that every Menger space can be regarded as a fuzzy metric space, that with a weak condition on the right bounding function a fuzzy metric space induces a metrizable uniformity on the underlying set, and that certain fixed point theorems hold.

Reviewer: A.J.Klein

##### MSC:

54A40 | Fuzzy topology |

54E35 | Metric spaces, metrizability |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

##### Keywords:

statistical metric space; fuzzy numbers; Menger space; bounding functions; fuzzy metric space
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\textit{O. Kaleva} and \textit{S. Seikkala}, Fuzzy Sets Syst. 12, 215--229 (1984; Zbl 0558.54003)

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