Characterizations of some fuzzy topological notions in probabilistic metric spaces

Abstract

Abstract Since Hohle's work in 1982 on the probabilistic metrization of fuzzy uniformities, Menger's probabilistic metric spaces (considered here under the triangular norm Min only) have been endowed with a number of fuzzy topological structures. We introduce ‘topology-free’ characterizations of the three so far known types of continuity there; directly in terms of probabilistic metrics. We show that the associated level topologies are pseudo-metrizable, and in particular that a fuzzy neighbourhood space is probabilistic pseudo-metrizable iff its α-level topologies, for all 0