Ditopological texture spaces and fuzzy topology, I. Basic concepts

This is the first of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. The authors begin by defining the notion of q-sets, which together with the p-sets considered earlier, enable the formulation of a powerful concept of duality. This plays an important role in the theory of direlations and difunctions, which is described here in detail. Difunctions are then taken as the morphisms of a category dfTex, whose objects are texture spaces. Several important subcategories are defined and the closely related construct fTex defined. Some properties of the functors between these categories are obtained.