FUZZY METRIC NEIGHBORHOOD SPACES

Hohle founded the first coherent topological study of Menger's probabilistic metric spaces by endowing them with fuzzy T-uniformities. We continue Hohle'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 introduce 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). Using this N-Euclidean space, we introduce an axiom of N-complete regularity which characterizes the Lowen fuzzy uniform topologies.