Fuzzy relation-preserving maps and regular fuzzy topological spaces

Investigations of situations in which the existence of relations between fuzzy topological spaces implies the existence of functions between them are of importance because they carry the similarity between the spaces from the empirical to the structural realm. This paper explores the possibility that weak structural (or quasi-empirical) fuzzy relationships may be rendered structural by placing restrictions of certain kinds on the overall structure(s) of the underlying fuzzy topological spaces; specifically, an attempt is made to build continuous functions from subcontinuous ones. It is shown that if the domain of the relation is constrained to be fuzzy Hausdorff, if the range of the relation is constrained to be fuzzy regular, and if certain other fairly straightforward conditions are met, it is indeed possible to derive a continuous function from an underlying subcontinuous one. It is possible that such derived continuous functions may have implications relative to the solution of fuzzy relational equations and in other areas of fuzzy logic as well.