Characterizations Of Coverings For Upper Approximation Operators To Be Closure Operators

In this paper, we give characterizations of covering for two types of the upperapproximation operators to be closure operators. These results partially answer anopen problem on characterizations of covering for upper approximation operatorsto be closure operators which was raised in the paper [7]. The main results are thefollowings:Theorem3.1.1For a covering-based approximation space (U, C), C4is a clo-sure operator if and only if for any x, y∈U, if there exist C∈C, such that x∈Cand y∈/C, then exists C′∈C, such that y∈C′and x∈/C′.Theorem3.1.2For a covering-based approximation space (U, C), C4is a clo-sure operator if and only if there exists a T1topology τ on U, such that C is asubbase for (U, τ).Theorem3.1.3For a covering-based approximation space (U, C), C4is a clo-sure operator if and only if approximation space (U, C) can be described as such aninformation exchange system that for any two members x, y in this system, if x hassome information can not be shared with y, then y also has some information cannot be shared with x.Theorem3.1.4For a covering-based approximation space (U, C), C4is a clo-sure operator if and only if for any x, y∈U, the restriction C{|x, y}of C on {x, y}is not a monotone covering.Theorem3.2.1For a covering-based approximation space (U, C), C5is a clo-sure operator if and only if for any x, y∈U, if there exists p∈U such that p∈Cfor any C∈C with x∈C or y∈C, then there exist z∈U such that x∈C′andy∈C′for any C′∈C with z∈C′.Theorem3.2.2For a covering-based approximation space (U, C), C5is a clo-sure operator if and only if the family N={N (x): x∈U} is a base of a topology τ on U satisfying that for each N∈N, there exists N∈N such that Nis clopenin the topology space (U, τ) and N′N.Theorem3.2.3For a covering-based approximation space (U, C), C5is a clo-sure operator if and only if approximation space (U, C) can be described as such aninformation exchange system that for any two members x, y in this system, if thereexists a member p such that p can share all the information owned by x or y, thenthere exists a member z such that both x and y can share all the information ownedby z.