Characterizations of coverings for upper approximation operators being closure operators ?
In this paper, we define two conditions of symmetry for the covering C C 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 C C 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.
closure operator triangle chain condition covering-based upper approximation operator first symmetrical condition second symmetrical condition