Maps between covering approximation spaces and the product space of two covering approximation spaces

Abstract

Let (V,D) be a covering approximation space and let f:U⟶V be a surjective map, where U is a nonempty set. We can achieve a covering C of U by defining C={f−1(D):D∈D}. This is a natural way to construct a covering for U and this motivates the research of covering rough sets from the mapping point of view. In this paper, we have shown that the inverse image of a neighborhood under f is a neighborhood and the inverse image of the closure of a point under f is again the closure of the corresponding point. Thus some approximation operators on the two covering approximation spaces must be related to each other. Some theorems and propositions with respect to the relationships between the approximation operators defined on U and the corresponding ones defined on V have been obtained. The purpose is to emphasize the importance of mappings between two covering approximation spaces. Finally, given two covering approximation spaces (U1,A) and (U2,B), we define the product covering for U1×U2 as {A×B|A∈A,B∈B}. This is a method to construct a new covering approximation space out of existing ones. Then we research the connections between the neighborhoods and closures of U1×U2 and the corresponding ones of factor spaces. In the last part of this paper, we explore the connections between the approximation operators defined on U1×U2 and the corresponding ones defined on factor spaces.