Fuzzy metric neighbourhood spaces (Q1189915)

This paper is a continuation of Höhle's work on probabilistic metrization of fuzzy uniformities [\textit{U. Höhle}, ibid. 8, 63-69 (1982; Zbl 0494.54006)]. Höhle's fuzzy uniformity (fuzzy \(T\)- uniformity, where \(T\) is a continuous \(t\)-norm) coincides with the Lowen fuzzy uniformity when \(T=\text{Min}\). The probabilistic metric as considered here induces a min-fuzzy uniformity which in turn induces a fuzzy neighborhood space [\textit{R. Lowen}, ibid. 7, 165-189 (1982; Zbl 0487.54008)]. The authors begin with this fuzzy neighborhood space which they termed as fuzzy metric neighborhood space, and developed quite a large number of results on it. The authors also introduce \(N\)-Euclidean space which serves as an example of fuzzy metric neighborhood space [see also \textit{U. Höhle}, ibid. 1, 311-332 (1978; Zbl 0413.54002)]. Using Artico-Moresco fuzzy proximities [\textit{G. Artico} and \textit{R. Moresco}, ibid. 21, 85-98 (1987; Zbl 0612.54006)] the authors introduce fuzzy metric proximities. Separation axioms such as \(N\)-regularity, \(N\)- normality and \(N\)-complete regularity, in addition to [\textit{P. Wuyts} and \textit{R. Lowen}, J. Math. Anal. Appl. 93, 27-41 (1983; Zbl 0515.54004)] lower separation axioms are studied in this fuzzy metric neighborhood space.