Powerset Operators Based Approach To Fuzzy Topologies On Fuzzy Sets

Topological structures have been extensively studied in the context of fuzzy set theories, and many and well organized approaches to this matter have been developed (see [7]). This chapter merges and extends two notions of fuzzy topological spaces, each of which extends the Chang-Goguen approach to fuzzy topology: the first notion considers a fuzzy topological structure as a suitable fuzzy set on some set (see [10,11,15]); the second notion introduces fuzzy topologies on (not necessarily crisp) fuzzy sets (see [2,3,5,9]). Indeed we can speak here of fuzzy topological spaces on fuzzy sets since both the carrier and the topology of the space are fuzzy sets. As a ground category we consider the category of fuzzy sets on a fixed base, already introduced in [3,4,5]. The category of fuzzy topological spaces described here is topological over its ground. In such a category the fundamental concepts of subspaces and product spaces are also introduced. But the main character of the present chapter lies in considering and using fuzzy powerset operators in order to give a description of the presented matter. Such operators were introduced and studied, in somehow different context, in [12,13] as well as in [3,5]. Here fuzzy powerset operators allow us to simplify notation and to enlighten quite well the role of distributivity conditions in the used lattices.