arXiv subjects

Xiaoyong Xi

Publications and source records attributed to Xiaoyong Xi.

23 records · Page 2Linked to original sources

arXiv2019

First countability, $ω$-well-filtered spaces and reflections

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.

math.GN↗

arXiv2019

On $T_0$ spaces determined by well-filtered spaces

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.

math.GN↗

arXiv2019

Existence of well-filterifications of $T_0$ topological spaces

We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $η_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous mapping $\hat{f}: W(X)\lra Y$ such that $f=\hat{f}\circ η_X$. Such a space $W(X)$ will be called the well-filterification of $X$. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.

math.GN↗

arXiv2019

A complete Heyting algebra whose Scott space is non-sober

We prove that (1) for any complete lattice $L$, the set $\mathcal{D}(L)$ of all nonempty saturated compact subsets of the Scott space of $L$ is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice $L$ is non-sober, then the Scott space of $\mathcal{D}(L)$ is non-sober. Using these results and the Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus give a positive answer to a problem posed by Jung. We will also prove that a $T_0$ space is well-filtered iff its upper space (the set $\mathcal{D}(X)$ of all nonempty saturated compact subsets of $X$ equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.

math.GN↗

arXiv2018

The Ho-Zhao Problem

Given a poset $P$, the set, $Γ(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $Γ$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $Γ(P) \cong Γ(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $Γ$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $Γ$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $Γ$-faithful. This subcategory subsumes all currently known $Γ$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

cs.LO↗