Fuzzy [formula omitted]-neighbourhood spaces: Part 2— [formula omitted]-neighbourhood systems

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.