Spaces Which are Dually Special Property \mathscrP and Related Conclusions

Let X be a space,and \mathscrP=A∶A is a subset of X, and has property >\mathscrP.A space X is dual the property \mathscrP if for any neighborhood assignment φ for X,there is a subset A⊂X,A∈\mathscrP,such that X=∪φ(x)∶x∈A.In this note,we mainly discuss properties of spaces which are dually special \mathscrP,and also give a necessary and sufficient condition for spaces which are dually special \mathscrP.These conclusions can be held for many spaces.As a corollary,we have that if X is a regular weak θ-refinable(dually discrete)-scattered space,then X is dually discrete．We also get some conclusions conserning aD- spaces．