Fuzzy T-neighbourhood spaces: Part 1 - T-proximities

This is the 5rst one of three articles that are intended to be published consecutively in this journal. Together they constitute a three-part work on the uni5ed subject of fuzzy T -neighbourhood spaces and their fuzzy T -proximities, where T stands for any continuous triangular norm. These new notions generalize, to arbitrary T , corresponding ones due to Artico, H8 ohle, Lowen and Moresco. In this part, we de5ne and study notions of fuzzy T -proximity, one for each T . Their de5nition subsumes that of fuzzy proximity due to Artico and Moresco (Fuzzy Sets and Systems 21 (1987) 85), as our Min-proximity. In particular, we study proximity maps, the fuzzy topological space associated with a T -proximity (which, in the second part, will be proved a fuzzy T -neighbourhood space), T -proximities as fuzzy relations in (ordinary) power sets, T -proximal neighbourhood systems, and relationships to the H8 ohle–Lowen fuzzy T -uniformities (J. Math. Anal. Appl. 82 (1981) 370; Manuscripta Math. 38 (1982) 289). The second part of this three-part work will be concerned with the categories of fuzzy T -neighbourhood spaces, one for each T . Whereas we devote the third part to our T -separation axioms in those categories. c © 2002 Elsevier Science B.V. All rights reserved.