Abstract We first introduce and study two new classes of subsets in T 0 spaces — ω-Rudin sets and ω-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces — ω-d-spaces and ω-well-filtered spaces. We prove that an ω-well-filtered T 0 space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable ω-well-filtered T 0 space are directed. Therefore, a first countable T 0 space X is sober iff X is well-filtered iff X is an ω-well-filtered d-space. Using ω-well-filtered determined sets, we present a direct construction of the ω-well-filtered reflections of T 0 spaces, and show that products of ω-well-filtered spaces are ω-well-filtered.

中文翻译：

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第一可数性，ω 良好过滤的空间和反射

摘要 我们首先介绍并研究了 T 0 空间中两类新的子集——ω-Rudin 集和 ω-well-filtered 确定集，它们位于可数有向子集的所有闭包类和不可约闭子集的类之间，以及两种新类型空间 - ω-d-空间和 ω-过滤良好的空间。我们证明一个 ω 良好过滤的 T 0 空间是局部紧的，如果它是核心紧致的。一个直接的推论是，每个核心紧凑的过滤良好的空间都是清醒的，用一种新方法回答了嘉荣问题。我们还证明了第一个可数 ω-well-filtered T 0 空间中的所有不可约闭合子集都是有向的。因此，第一个可数的 T 0 空间 X 是清醒的，如果 X 是过滤良好的，如果 X 是 ω 良好过滤的 d 空间。使用 ω 良好过滤的确定集，