Topological structures of interval-valued hesitant fuzzy rough set and its application

Abstract

Interval-valued hesitant fuzzy rough set, defined by Zhang et al. (55), is an extension of hesitant fuzzy rough sets, interval- valued fuzzy rough sets and fuzzy rough sets. For further studying the theories and applications of interval-valued hesitant fuzzy rough sets, in this paper, we mainly investigate the topological structures of interval-valued hesitant fuzzy rough sets. Firstly, the concept of interval-valued hesitant fuzzy topological spaces is introduced by us. Then relationships between interval-valued hesitant fuzzy rough approximation spaces and interval-valued hesitant fuzzy topological spaces are further established. It is proved that the set of all lower approximation sets based on an interval-valued hesitant fuzzy reflexive and transitive approximation space forms an interval-valued hesitant fuzzy topology; and conversely, for an interval-valued hesitant fuzzy rough topological space, there exists an interval-valued hesitant fuzzy reflexive and transitive approximation space such that the topology in the interval- valued hesitant fuzzy rough topological space is just the set of all lower approximation sets in the interval-valued hesitant fuzzy reflexive and transitive approximation space. That is to say, there exists a one-to-one correspondence between the set of all interval-valued hesitant fuzzy reflexive and transitive approximation spaces and the set of all interval-valued hesitant fuzzy rough topological spaces. Finally, a practical application is provided to illustrate the validity of the interval-valued hesitant fuzzy rough set model.