Topological structures of a type of granule based covering rough sets

Abstract

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindelöf property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

Numerical Characterizations of Topological Reductions of Covering Information Systems in Evidence Theory

“…Proposition 1 (see [7,18,53,54]). Let C be a covering of the universe U. en, for any X, Y ⊆ U, we get (1)…”

“…Corollary 1 (see [40,54]). Let C be a covering of the universe U. en, (1) FL C and LL C are interior operators.…”