On Fuzzy Topological Structures of Rough Fuzzy Sets

Abstract

A rough fuzzy set is the result of approximation of a fuzzy set with respect to a crisp approximation space. In this paper, we investigate topological structures of rough fuzzy sets. We first show that a reflexive crisp rough approximation space can induce a fuzzy Alexandrov space. We then prove that the lower and upper rough fuzzy approximation operators are, respectively, the fuzzy interior operator and fuzzy closure operator if and only if the binary relation in the crisp approximation space is reflexive and transitive. We also examine that a similarity crisp approximation space can produce a fuzzy clopen topological space. Finally, we present the sufficient and necessary conditions that a fuzzy interior (closure, respectively) operator derived from a fuzzy topological space can associate with a reflexive and transitive crisp relation such that the induced lower (upper, respectively) rough fuzzy approximation operator is exactly the fuzzy interior (closure, respectively) operator.