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

We explore a notion of fuzzy T -neighbourhood spaces, for any continuous triangular norm T , and we present on this notion a uni6ed 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 H; ohle (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–H; ohle 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. c © 2002 Elsevier Science B.V. All rights reserved.