Fuzzy topological spaces

Abstract

We start by discussing a class of special fuzzy topological spaces, that is, the product-induced spaces [S]. First, we show that every fuzzy topological space is topologically isomorphic with a certain topological space, and then proceed to prove every open fuzzy set is defined by some lower semicontinuous function. Taking this as the background, we introduce the concept of dual points [6]. and thus establish a kind of neighborhood structure of fuzzy points such that the Q-neighborhood [4], which is one of the important notions in fuzzy topology, and the neighborhood arc integrated in this structure. This neighborhood structure will be the core of our developing the theory of fuzzy topological spaces. We introduce the concept of strong quasi-discoincident [ 1 I]. and so give a group of fuzzy separation properties which is a most natural generalization of the usual separation properties. Next, we introduce a kind of fuzzy metrics and use this metrics directly to discuss the fuzzy metric space. By means of fuzzy points, WC define kinds of uniformities, discuss their fundamental properties and extend Weil’s theorem on usual topology to fuzzy topological spaces, and hence obtain their separation character. Naturally, these fuzzy uniformities can still be characterized by a family of fuzzy metrics. Finally. we discuss the problem of fuzzy mctrization on fuzzy topological spaces and obtain a fuzzy metrization theorem which contains the Nagata-Smirnov theorem as a special example.