Fuzzy T-neighbourhood spaces: part 3: T-separation axioms

  • Authors:

  • Khaled A. Hashem;Nehad N. Morsi

  • Affiliations:

  • Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt;Department of Basic Sciences, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Miami, Alexandria, Egypt

  • Venue:
  • Fuzzy Sets and Systems - Topology
  • Year:
  • 2003

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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 T0 and R0, 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).