Fuzzy T-neighbourhood spaces. Part 3: T-separation axioms

Abstract

We address the problem of identifying a useful set of mutually compatible separation axioms for each category T-FNS of T-neighbourhood spaces. It should contain one of T-complete regularity that characterizes T-uniformizability, as indeed we achieve here. This seems to necessitate the parameterization of most separation axioms by the triangular norm T, with the exception of the two axioms T 0 and R 0, which have categorical definitions valid in any topological category. We introduce T-separation axioms in terms of T and fuzzy closure operators only; making them meaningful for the whole of FTS. We then characterize them within T-FNS in terms of T, fuzzy closures of crisp fuzzy subsets and T-neighbourhoods. We study our T-axioms and establish some entailments between them. Our T-complete regularity commits us to investigate a “T-real line”, which is a T-neighbourhood space induced by a certain probabilistic T-metric due to Höhle (Fuzzy Sets and Systems 1 (1978) 311).