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

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 T0 and R0, which have categorical de4nitions 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 H9 ohle (Fuzzy Sets and Systems 1 (1978) 311). c © 2002 Elsevier Science B.V. All rights reserved.