Topological characterizations of covering for special covering-based upper approximation operators

The relationships among properties of covering approximation operators and their corresponding coverings have attracted intensive research in recent years. In particular, those among topological properties have drawn special attention because of their important applications in rough set theory. In this paper, we give topological characterizations of covering C for covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C, and describe covering-based approximation space (U, C) as certain types of information exchange systems when SH or RH is a closure operator. By applying our new characterizations, we give inequalities about the relationship between the number of members in C and the number of elements in U, and discuss relationships among conditions for different covering based upper approximation operators to be closure operators. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Furthermore, in this paper we also give several characterizations of unary coverings, an important type of coverings in studying relationships among basic concepts in covering-based rough sets, by the relationships among different types of covering-base approximation operators.