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- W2973809529 abstract "We first introduce and study two new classes of subsets in $T_0$ spaces - Rudin sets and $wdd$ 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 - $mathsf{DC}$ spaces, Rudin spaces and $wdd$ spaces. The class of Rudin spaces lie between the class of $wdd$ spaces and that of $dc$ spaces, while the class of $dc$ spaces lies between the class of Rudin spaces and that of sober spaces. Using Rudin sets and $wdd$ 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 if{}f $X$ is a well-filtered Rudin space if{}f $X$ is a well-filtered $mathsf{WD}$ space. We also prove that every locally compact $T_0$ space is a Rudin space, and every core compact $T_0$ space is a $wdd$ space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using $wdd$ sets, we present a more directed 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 answering will deepen our understanding of the related spaces and structures." @default.
- W2973809529 created "2019-09-26" @default.
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- W2973809529 date "2019-09-19" @default.
- W2973809529 modified "2023-09-27" @default.
- W2973809529 title "On $T_0$ spaces determined by well-filtered spaces" @default.
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- W2973809529 doi "https://doi.org/10.48550/arxiv.1909.09303" @default.
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