Fuzzy grills and a characterization of fuzzy proximity

Abstract

In his classical paper [lo], Zadeh introduced the fundamental concept of fuzzy sets. This idea was then applied to various other fields of mathematics including topology resulting in the study of fuzzy topology by Chang [I], fuzzy uniformity by Lowen [7], and fuzzy proximity by Katsaras [5]. Thron IS] presented a new approach to poximity structures using the theory of grills, an important concept introduced by Choquet [3]. The importance of grills in the proximity theory is noticed sufficiently in various other papers (e.g., cf. [2,4]) also. Recently, Katsaras [6] introduced and studied filters, ultrafilters, and clusters in the new setup of fuzzy proximity on a set. In the present paper, we introduce fuzzy stacks and fuzzy grills and obtain a characterization of fuzzy proximity using fuzzy grills. In Section 2, fuzzy stacks and fuzzy grills and in Section 3 on the lines of basic proximity or C?-proximity as described in [8,9], the concept of fuzzy basic proximity on a set are introduced. Also, the fuzzy basic proximal neighborhood of a fuzzy set with respect to a fuzzy basic proximity is defined and some of the properties of sets of all fuzzy basic proximal neighborhoods of fuzzy sets are obtained. Finally in Section 4, a necessary and sufficient condition for a fuzzy basic proximity to be a fuzzy proximity is obtained.