Study of Covering Based Multi Granular Rough Sets and Their Topological Properties

The notions of basic rough sets introduced by Pawlak\udas a model of uncertainty, which depends upon a single\udequivalence relation has been extended in many directions. Over\udthe years, several extensions to this rough set model have been\udproposed to improve its modeling capabilities. From the\udgranular computing point of view these models are single\udgranulations only. This single granulation model has been\udextended to multi-granulation set up by taking more than one\udequivalence relations simultaneously. This led to the notions of\udoptimistic and pessimistic multi-granulation. One direction of\udextension of the basic rough set model is dependent upon covers\udof universes instead of partitions and has better modeling power\udas in many real life scenario objects cannot be grouped into\udpartitions but into covers, which are general notions of\udpartitions. So, multigranulations basing on covers called\udcovering based multi-granulation rough sets (CBMGRS) were\udintroduced. In the literature four types of CBMGRSs have been\udintroduced. The first two types of CBMGRS are based on\udminimal descriptor and the other two are based on maximal\uddescriptor. In this paper all these four types of CBMGRS are\udstudied from their topological characterizations point of view. It\udis well known that there are four kinds of basic rough sets from\udthe topological characterisation point of view. We introduce\udsimilar characterisation for CBMGRSs and obtained the kinds\udof the complement, union, and intersection of such sets. These\udresults along with the accuracy measures of CBMGRSs are\udsupposed to be applicable in real life situations. We provide\udproofs and counter examples as per the necessity of the\udsituations to establish our claims.\u