Notes about some point sets in topological space

Firstly,this article gives a sufficient and necessary condition when a set is a closed one in a topological space,than gets a closure in a topological space and a closed set whose closure is the smallest of all closed sets that contain it when the border set must be a closed one.Secondly,we knew that a derived set must be a closed one in the general metric space,but this conclusion is not necessarily proper in the general topological space,therefore this article mainly gives a sufficient and necessary condition when the derived set is a closed one in the topological space.