Abstract
This is the first one of three articles that are intended to be published consecutively in this journal. Together they constitute a three-part work on the unified 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, Höhle, Lowen and Moresco. In this part, we define and study notions of fuzzy T-proximity, one for each T. Their definition 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 Höhle–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.
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