Characterizations of coverings for upper approximation operators being closure operators

In this paper, we define two conditions of symmetry for the covering CC in a covering-based approximation space (U,C)(U,C). By using these conditions and the triangle chain condition, we give general, topological and intuitive characterizations of the covering CC for three types of covering-based upper approximation operators being closure operators. We also give descriptions of (U,C)(U,C) in terms of information exchange systems when these operators are closure operators. These results answer an open problem raised in Ge et al. (2012). As an example application of our characterizations, we discuss the problem of implications among conditions for different types of covering-based upper approximation operators being closure operators.