Fuzzy T-locality spaces

Abstract

A Lowen fuzzy topological space is said to be localizable at points if it is completely determined (in some prescribed manner) by a system of fuzzy neighbourhoods (called here “fuzzy localities”) of its ordinary points. We introduce, for each lower semicontinuous triangular norm T, a new type of those spaces, which we call fuzzy T-locality spaces. The particularity of a fuzzy T-locality space is that its fuzzy interior operator is obtained from its “ T-locality system” through a definition of fuzzy containment of fuzzy sets that employs the T-residuated implication operator. The origins of this new theory lie in the theory of probabilistic metric spaces. Indeed, fuzzy T-locality spaces provide a fuzzy topological framework for a naturally defined notion of “metric T-continuity” of functions between probabilistic T-metric spaces; in a manner described in the last two sections of this article. Before that, we establish characterizations and aspects of good behaviour for fuzzy T-locality spaces, including that every T-locality system has an open basis. We briefly study their level topologies, and we show that all topologically generated spaces are fuzzy T-locality spaces. We prove that the category T-FLS, of fuzzy T-locality spaces and continuous functions between them, is a topological category. We show that it is closed in FTS under the formation of optimal lifts of sources. We provide a condition on T, under which T-FLS becomes closed in FTS also under the formation of co-optimal lifts of sinks.