关于可展空间的完备原象

ON PERFECT PREIMAGE OF DEVELOPABLE SPACES

本文证明:一个空间X是一个T_1可展空间的拟完备原象当且仅当X中存在开覆盖的核正规序列{n:n∈N},满足以下两个条件:(1)对每一x∈X,c(x)=st(x,n)是闭集,(2)如x_n∈st(x,n),n∈N,则{x_n}在c(x)中有聚点。此外,还指出T_1可展空间的完备原象具有如下的覆盖性质:它的每个定向开覆盖都有可剖的开加细。

We prove that (1)space X is a quasi-perfect preimage of a T1 developable space iff there exists a kernel-normal sequence of open covers of X such that (i) foreach x∪X,c(x)= st(x)is closed (ii) if xn∪st(x,) ,then {xn} has an accumulationpoint in c(x),(2)Any perfect preimage of a T1 clevelopabel space is weakly D-paracompact.