Fuzzy T-neighbourhood spaces: part 2--T-neighbourhood systems

  • 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 - Mathematics
  • Year:
  • 2002

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Abstract

We explore a notion of fuzzy T-neighbourhood spaces, for any continuous triangular norm T, and we present on this notion a unified treatment. Our theory, on one hand, generalizes the theory of Lowen (Fuzzy Sets and Systems 7 (1982) 65) from T = Min to arbitrary T, which has been the progenitor of this work, and on the other hand it is strongly related to the theory of L-neighbourhoods of Höhle (Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht, 1999) (when L is restricted to the lattice [0, 1]). We study the T-neighbourhood bases and systems of our spaces, their level topologies, as well as their relationship to the Lowen-Höhle fuzzy T-uniformities (J. Math. Anal. Appl. 82 (1981) 370; Manuscripta Math. 38 (1982) 289) and to our fuzzy T-proximities (introduced in Part 1). We show that the nearness concept underlying a fuzzy T-neighbourhood space is a fuzzy relation of closeness between its ordinary subsets and ordinary points. In particular, our spaces are in canonical one-to-one correspondence with the (T-)probabilistic topological spaces of Frank (J. Math. Anal. Appl. 34 (1971) 67). We demonstrate that each full subcategory T-FNS, of FTS, of fuzzy T-neighbourhood spaces is a topological category. To do that, we characterize |T-FNS| within |FTS|, and we characterize continuity of functions within T-FNS in terms of T-neighbourhood bases.