Fuzzy metric neighbourhood spaces

Abstract

Höhle founded the first coherent topological study of Menger's probabilistic metric spaces by endowing them with fuzzy T-uniformities. We continue Höhle's work in the special case when the triangular norm T used is Min, that is when the resulting probabilistic metric uniformities are Lowen fuzzy uniformities. In this case, we call the induced fuzzy topological spaces fuzzy pseudo-metric neighbourhood spaces. We introduce axioms of N-regularity and N-normality, which are satisfied in fuzzy pseudo-metric neighbourhood spaces. We introduced a Lowen fuzzy uniform topology on the fuzzy real line R I, called the N-Euclidean topology, by means of a fuzzy (=probabilistic) metric on R( I), called the axiom of N-complete regularity which characterizes the Lowen fuzzy uniform topologies.