On T0 spaces determined by well-filtered spaces
On T0 spaces determined by well-filtered spaces
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在由经过良好过滤的空间确定的 T0 空间上
DOI：
10.1016/j.topol.2020.107323
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发表时间：
2020
影响因子：
0.6
通讯作者：
Dongsheng Zhao
中科院分区：
文献类型：
--
作者：
Xiaoquan Xu;Chong Shen;Xiaoyong Xi;Dongsheng Zhao
首先在T0空间中引入并研究了两类新的子集--Rudin集和WD集，它们位于有向子集的所有闭包和不可约闭子集的闭包之间。利用这些子集，我们定义了三种新的拓扑空间-DC空间、Rudin空间和WD空间。Rudin空间介于WD空间和DC空间之间，DC空间介于Rudin空间和清醒空间之间。利用Rudin集和WD集，我们给出并证明了良滤子空间和清醒空间的一些新的刻画。对于T0空间X，证明了X是清醒的当且仅当X是良滤鲁丁空间当且仅当X是良滤子WD空间。证明了每个局部紧T0空间是Rudin空间，每个核紧T0空间是WD空间。一个直接的推论是，每一个核心紧凑、过滤良好的空间都是清醒的，给出了对佳荣问题的肯定答案。利用WD集更直接地构造了T0空间的良滤性反射，并证明了任何良滤性空间的乘积都是良滤性的。我们的研究也引出了一些问题，这些问题的答案将加深我们对相关空间和结构的理解。
We first introduce and study two new classes of subsets in T 0 spaces—Rudin sets and WD sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, we define three new types of topological spaces—DC spaces, Rudin spaces and WD spaces. Rudin spaces lie between WD spaces and DC spaces, while DC spaces lie between Rudin spaces and sober spaces. Using Rudin sets and WD sets, we formulate and prove a number of new characterizations of well-filtered spaces and sober spaces. For a T 0 space X, it is proved that X is sober iff X is a well-filtered Rudin space iff X is a well-filtered WD space. We also prove that every locally compact T 0 space is a Rudin space, and every core compact T 0 space is a WD space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using WD sets, we present a more direct construction of the well-filtered reflections of T 0 spaces, and prove that the products of any collection of well-filtered spaces are well-filtered. Our study also leads to a number of problems, whose answers will deepen our understanding of the related spaces and structures.