On an Open Problem for Definable Subsets of Covering Approximation Spaces

Abstract

Let (U ;D) be a Gr-covering approximation space (U ; C) with covering lower approximation operator D and covering upper approximation operator D. For a subset X of U , this paper investigates the following three conditions: (1)X is a definable subset of (U ;D); (2) X is an inner definable subset of (U ;D); (3) X is an outer definable subset of (U ;D). It is proved that if one of the above three conditions holds, then the others hold. These results give a positive answer of an open problem for definable subsets of covering approximation spaces. Keywords—Covering approximation space, covering approximation operator, definable subset, inner definable subset, outer definable subset.