不具有性质(wa)的拓扑空间

Topological Spaces Without Property (wa)

设（X,T）是拓扑空间,如果对于任意的开覆盖U和任意的稠密子集D,存在X的离散子集F（?）D使得St（F,U）=U{U∈U:U∩F≠（?）}=X,则称（X,T）具有性质（wa）.每一正规空间都具有性质（wa）.M.V.Matveev举例说明了T1空间可以不具有性质（wa）.本文证明了存在很多Hausdorff空间不具有性质（wa）,且进一步举例说明了Tychonoff空间可以不具有性质（wa）,这些结果回答了Matveev的问题.

A topological space (X,T) has property (wa) (property (a), respectively) provided for every open cover U of (X, T) and every dense set D there exists a subset F C D such that F is discrete (closed and discrete, respectively) and St(F,U) = U{U U : U F =0} = X. M.V. Matveev gave a T1 space without property (wa). In the present paper, a method constructing Hausdorff spaces without property (wa) is given. Moreover, a Tychonoff space without property (wa) is obtained. As a corollary, we answer a question of M.V. Matveev.