Digital topological rough set structures and topological operators

Abstract

The present paper studies digital topological rough set structures associated with Khalimsky (K-, for brevity) topology. We further explore their properties related to topological (closure and interior) operators. In addition, we find that the K-topological rough set structures have their own features different from certain theoretical properties of the H-rough set structure in [9]w.r.t. the topological operators. More precisely, whereas the H-rough set structure is a topological operator, K-topological concept approximations are proved not to be topological operators. Thus, this paper proposes an alternative approach to the K-topological concept approximations so that we finally have new types of topological operators. Furthermore, it appears that this new ones are smoothly matched with the earlier H-rough set structure from the viewpoint of topological operators. Therefore, they can facilitate the study of continuous-oriented or digital-oriented rough sets, which can be used in applied sciences without any limitations of the cardinalities of the universe and a target set. Besides, they can support certain decision rules with no restriction to anything continuous or discrete (or digital). In addition, we introduce the notion of K-definability of a target subset of a universal set and write an algorithm for estimating the roughness of the above new concept approximations. In this paper each set need not be finite and a covering is locally finite.