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In this paper, we discuss the approximation operatorsa  p  r   ¯    N  S${\overline {apr} _{NS}}$anda  p  r   ¯   S${\overline {apr} _S}$which are based on  NS ( U ) and  S . We not only obtain some properties of  NS ( U ) and  S , but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

The characterizations of upper approximation operators based on special coverings