可数-定向集与ω^(*)-well-filtered空间

Countably-directed Sets and ω^(*)-well-filtered Spaces

本文利用可数-定向集,引入了ω^(*)-Rudin空间和ω^(*)-well-filtered决定空间的概念,讨论了ω^(*)-well-filtered空间及相关空间的收缩性、乘积性、反射性等基本性质,给出了ω^(*)-well-filtered性的等价刻画,得到了如下主要结果:(1)ω^(*)-Rudin空间的连续收缩核仍然是ω^(*)-Rudin空间;(2)ω^(*)-well-filtered决定空间的连续收缩核仍然是ω^(*)-well-filtered决定空间;(3)设{X_(i):1≤i≤n}是有限个T_(0)空间,则n∏i=1X_(i)是ω^(*)-well-filtered决定空间当且仅当每个因子空间X_(i)是ω^(*)-well-filtered决定空间(i=1,2,…,n);(4)ω^(*)-well-filtered空间的连续收缩核是ω^(*)-well-filtered空间;(5)设{X_(i):i∈I}是一族T_(0)空间,则∏i∈IX_(i)是ω^(*)-well-filtered空间当且仅当每个因子空间X_(i)是ω^(*)-well-filtered空间(i∈I);(6)ω^(*)-well-filtered空间范畴是T_(0)空间范畴的满子反射范畴,并给出了T_(0)空间ω^(*)-well-filtered反射的具体构造。

In this paper,using countably-directed sets,we introduce the concepts of ω^(*)-Rudin spaces and to ω^(*)-well-filtered determined spaces,and discuss some basic properties of ω^(*)-well-filtered spaces and related spaces.Some characterizations of ω^(*)-well-filteredness are given and the following results are obtained;(1)A retract of an ω^(*)-Rudin space is an ω^(*)-Rudin space;(2)A retract of an ω^(*)-well-filtered determined space is an ω^(*)-well-filtered determined space;(3)For a finite family{X_(i):1≤i≤n}of T_(0) spaces,the product space n∏i=1X_(i) is ω^(*)-well-filtered determined space if and only if for any i=1,2,•••n,X_(i)is ω^(*)-well-filtered i=i determined space;(4)The retract of an ω^(*)-well-filtered space is an ω^(*)-well-filtered space;(5)For a family{X_(i):i∈I}of T_(0) spaces,the product space ∏i∈IX_(i)is ω^(*)-well-filtered space if and only if for any i G I,X,is<e/ω^(*)-well-filtered space;(6)The category of all ω^(*)-well-filtered space is reflective in that of all T_(0) spaces,and the concrete ω^(*)-well-filtered reflective of a T_(0) space is given.