CATEGORIES OF FUZZY TOPOLOGY IN THE CONTEXT OF GRADED DITOPOLOGIES ON TEXTURES

Abstract

This paper extends the notion of ditopology to the case where openness and closedness are given in terms of a priori unrelated grading func- tions. The resulting notion of graded ditopology is considered both in the set- ting of lattices and in that textures, the relation between the two approaches being discussed in detail. Interrelations between graded ditopologies and di- topologies on textures are also studied. In classical topology the notion of open set is usually taken as primitive with that of closed set being auxiliary. However, since the closed sets are easily obtained as the complements of open sets they often play an important, sometimes dominating role in topological arguments. Much the same situation holds for topologies on lat- tices, where the role of set complement is played by an order reversing involution. It is the case, however, that there may be no order reversing involution available, or that the presence of such an involution is otherwise irrelevant to the topic un- der consideration. To deal with such cases it is natural to consider a topological structure consisting of a priori unrelated families of open sets and of closed sets. This was the approach adapted from the beginning for the topological structures on textures, originally introduced as a point-based representation for fuzzy sets (2, 3). These topological structures were given the name dichotomous topology, or ditopology for short. They consist of a family of open sets and a generally unrelated family of closed sets. Hence, both the open and the closed sets are regarded as primitive concepts for a ditopology. A ditopology (; ) on the discrete texture (X;P(X)) gives rise to a bitopological space (X;; c ). This link with bitopological spaces has had a powerful inuence on the development of the theory of ditopological texture spaces, but it should be emphasized that a ditopology and a bitopology are conceptually dierent. Indeed, a bitopology consists of two separate topological structures (complete with their open and closed sets) whose interrelations we wish to study, whereas a ditoplogy represents a single topological structure. There is no reason why the notion of ditopology should be restricted to textures, and indeed in two recent papers it has been extended to completely distributive