On the Relationship Between Several Basic Categories in Fuzzy Topology

Abstract

In this paper, a process, which associates with every fibre-small topo- logical category A and every completely distributive lattice L another topological category called the co-tower extension of A, is introduced to investigate the relationship between several basic categories in fuzzy topology. It is proved that the category of stratified Šostak L-fuzzy topological spaces is the co-tower extension of the category of stratified Chang-Goguen spaces; and the category of L-fuzzifying topological spaces (which is isomorphic the category of fuzzy neighborhood spaces when L = [0, 1]) is the co-tower extension of the category of topological spaces. Thus, in some sense, we can say that properties of the category of stratified Šostak L-fuzzy topological spaces and L-fuzzifying topological spaces are determined by the categories of stratified Chang-Goguen spaces, topological spaces and the lattice L. Therefore it can be said that in fuzzy topology the category of (stratified) Chang-Goguen topological spaces is more basic and more important than the other categories.