On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations

Topologies and rough set theory are widely used in the research field of machine learning and cybernetics. An intuitionistic fuzzy rough set, which is the result of approximation of an intuitionistic fuzzy set with respect to an intuitionistic fuzzy approximation space, is an extension of fuzzy rough sets. For further studying the theories and applications of intuitionistic fuzzy rough sets, in this paper, we investigate the topological structures of intuitionistic fuzzy rough sets. We show that an intuitionistic fuzzy rough approximation space can induce an intuitionistic fuzzy topological space in the sense of Lowen if and only if the intuitionistic fuzzy relation in the approximation space is reflexive and transitive. We also examine the sufficient and necessary conditions that an intuitionistic fuzzy topological space can be associated with an intuitionistic fuzzy reflexive and transitive relation such that the induced lower and upper intuitionistic fuzzy rough approximation operators are, respectively, the intuitionistic fuzzy interior and closure operators of the given topology.