Covering rough set structures for a locally finite covering approximation space

Abstract

The purpose of this paper is to introduce the notion of locally finite covering approximation (LFC-, for brevity) space (U, C) which is a generalization of a (finite) covering approximation space. Furthermore, for a subset X of the universe U, we develop two kinds of neighborhood systems derived from the covering C, specifically K1(X) and K2(X) which are not comparable with each other. Next, the paper establishes two topologies on U, Alexandroff, and quasi-discrete topologies generated by systems K1(X) and K2(X) as bases, respectively.After suggesting many examples explaining the above notions and systems, we develop two new types of rough set structures, an H-rough set operator (H*, H*) and a Khalimsky (K-, for short) topological rough set operator (DK−,DK+).The present paper investigates various properties of these operators and refers to certain advantages of them which can be used in applied science including digital geometry, information geometry, computer vision, pattern recognition, image processing, and so on. In order to accomplish this, we use some tools such as granulations, neighborhood systems, typical Pawlak’s tools, related topologies, rough set structures associated with the K-topology, and so on. Furthermore, through this work, we can obtain strong connections between covering rough set theory and applied geometry. In the present paper all sets U, C and X(⊆U) need not be finite.