Connectedness in ditopological texture spaces

Abstract

The notion of a texture space, under the name of fuzzy structure, was introduced by Brown (1993) as a means of representing a lattice of fuzzy sets as a lattice of crisp subsets of some base set. Corresponding to a fuzzy topology we then have the concept of a ditopology, and in this paper connectedness in ditopological texture spaces is defined, and some of its basic properties are obtained. In particular it is shown that connectedness is productive in the context of ditopological texture spaces. Zero-dimensionality and strong zero-dimensionality are defined, and it is shown that in stable zero-dimensional ditopological texture spaces the component of a point coincides with the quasi-component. Finally, it is shown that a fuzzy topological space is C 5 -connected if and only if the corresponding ditopological texture space is connected.