Some Categorical Aspects of Fuzzy Topology

Abstract

Research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties. In this paper we discuss some of these aspects. Since the topological construct L-FTS contains nontrivial both initially and finally closed full subconstructs, and each such construct gives rise to a natural autonomous theory of fuzzy topology, it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories, including the classical topology theory as a special case, with each applying to one such subconstruct. Therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of L-FTS to illustrate this idea. The second topic of this paper is the relationship between the construct of stratified L-fuzzy topological spaces and several other familiar constructs in fuzzy topology, for example, the constructs of Šostak fuzzy topological spaces and L-fuzzifying topological spaces.