Marcus–Wyse topological rough sets and their applications.

Abstract The aim of this paper is to establish two new types of rough set structures associated with the Marcus–Wyse (MW -, for brevity) topology, such as an M -rough set and an MW -topological rough set. The former focuses on studying the rough set theoretic tools for 2-dimensional Euclidean spaces and the latter contributes to the study of the rough set structures for digital spaces in Z 2 , where Z is the set of integers. These two rough set structures are related to each other via an M -digitization. Thus, these can successfully be used in the field of applied science, such as digital geometry, image processing, deep learning for recognizing digital images, and so on. For a locally finite covering approximation (LFC -, for short) space (U , C) and a subset X of U , we firstly introduce a new neighborhood system on U related to X. Next, we formulate the lower and upper approximations with respect to X , where all of the sets U and X (⊆ U) need not be finite and the covering C is locally finite. Actually, the notion of M -digitization of a 2-dimensional Euclidean space plays an important role in developing an M -rough and MW -topological rough set structures. Further, we prove that M -rough set operators have a duality between them. However, each of MW -topological rough set operators need not have the property as an interior or a closure from the viewpoint of MW -topology. [ABSTRACT FROM AUTHOR]