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Xiaoyong Xi

Publications and source records attributed to Xiaoyong Xi.

At least 19 recordsLinked to original sources
arXiv2026

Sobriety of Regular Open Algebras in Second-Countable T3 Spaces

For a topological space X, let RO(X) be the complete Boolean algebra of regular open subsets of X, ordered by inclusion. We prove that, for every second-countable T3 space X, the Scott space of RO(X) is sober if and only if the set of all isolated points of X is dense in X. Consequently, RO$(\mathbb R^n)$ is not sober for every positive integer $n$. In particular, RO$(\mathbb R)$ is not sober, which provides an answer to an open problem concerning the sobriety of complete Boolean algebras. This characterization also yields a systematic way to obtain more natural examples of complete lattices whose Scott spaces are non-sober.

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arXiv2026

A Quasicontinuous Domain Without an Interval Retract

We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binary tree and the binary-value map, a well-founded quasicontinuous dcpo \[ P=2^{<ω}\mathbin{\dot\cup}[0,1] \] which is not quasialgebraic. Moreover, $P$ contains no sub-dcpo isomorphic to $[0,1]$, and $[0,1]$ is not a Scott-continuous retract of $P$.

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arXiv2026

The answers to two problems on dcpo models

A poset model of a topological space $X$ is a poset $P$ such that $X$ is homeomorphic to the maximal point space of $P$ (the set Max($P$) of all maximal points of $P$ equipped with the relative Scott topology of $P$). Xi and Zhao proved that if a space has a dcpo model satisfying Lawson condition, it must be coherent and well-filtered. It is still open wether every coherent and well filtered space $T_1$ has a dcpo model satisfying Lawson condition. In this paper, we answer this problem. In another paper, Xi and Zhao proved that every Hausdorff k-space has a bounded complete dcpo model. It is, however, still unknown whether it is true that if a Hausdorff space is a k-space if it has a bounded complete dcpo model. We will construct a Hausdorff space which is not a k-space but has a bounded complete dcpo model.

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arXiv2026

Convergence Choquet-complete spaces and domain representations

de Brecht, Goubault-Larrecq, Jia and Lyu asked whether every sober convergence Choquet-complete space is domain-complete. We introduce the notion of singleton Choquet-completeness, a weakening of convergence Choquet-completeness in which the open sets chosen by player $α$ are required to have a singleton intersection, but not necessarily to form a neighbourhood basis. We prove that every singleton Choquet-complete $T_1$ space is domain-representable. Consequently, every convergence Choquet-complete $T_1$ space is domain-representable and hence sober. Thus, in the $T_1$ case, the sobriety assumption in the above question is redundant, and the question reduces to whether every convergence Choquet-complete $T_1$ space is domain-complete.

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arXiv2026

Scott Function Spaces under One-Sided FS Assumptions: Counterexamples, Positive Results, and New Directions

The class of FS-domains is known to be closed under Scott function spaces when both the source and target are FS-domains. This paper investigates what remains true under one-sided FS assumptions, with particular emphasis on the role of Plotkin's tie. We establish two complementary continuity theorems. First, whenever \(X\) is an FS-domain, the Scott function space \([X\to T]\) is a continuous dcpo. The proof introduces finite-layer truncation maps on Plotkin's tie, which generate directed families of way-below approximants below every Scott-continuous map. Secondly, whenever \(L\) is an FS-domain, the Scott function space \([T\to L]\) is again a continuous dcpo. Here the argument is based on finitely separating approximate identities, together with a finite-control analysis of the two-branch order structure of Plotkin's tie. These two approximation mechanisms are conceptually different but both produce the directed families of way-below approximants required for continuity. To determine the limits of these positive results, we consider the Lawson closed-disk domain. Although \(\Disk^{\top}\) is an FS-domain, the Scott function space \([\Disk^{\top}\to T]\) is shown to be continuous but not itself an FS-domain. This establishes that preservation of continuity is strictly weaker than preservation of the FS property. The paper concludes by identifying the boundaries of the present methods and proposing a unified approximation principle that may provide a general characterization of continuity for Scott function spaces.

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arXiv2026

Notes on countable frames

Matthew de Brecht raised the question of whether countable frames are continuous lattices. We prove that the continuity of a countable frame implies the quasicontinuity of its corresponding spectrum in the dual specialization order. We further show that this question admits a positive answer if the frame's spectrum is a $T_1$ space or a Scott space. In general, we confirm the existence of non-continuous countable frames. This work also partially addresses an open problem proposed by Jimmie Lawson and Michael Mislove in 1990, which concerns the characterization of when the spectrums of spatial frames are Scott spaces.

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arXiv2025

Products of two sober dcpo's need not be sober

We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober.

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arXiv2025

The answers to two problems on maximal point spaces of domains

A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ (with the relative Scott topology). We first construct an example to show that the set of maximal points of an ideal domain $P$ need not be a $G_δ$-set in the Scott space $ΣP$, thereby answering an open problem from Martin (2003). In addition, Bennett and Lutzer (2009) asked whether $X$ and $Y$ are domain-representable if their product space $X \times Y$ is domain-representable. This problem was first solved by Önal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem.

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arXiv2023

The set of maximal points of an $ω$-domain need not be a $G_δ$-set

A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$. Lawson proved that every Polish space $X$ has an $ω$-domain model $P$ and for such a model $P$, $\mbox{Max}(P)$ is a $G_δ$-set of the Scott space of $P$. Martin (2003) then asked whether it is true that for every $ω$-domain $Q$, $\mbox{Max}(Q)$ is $G_δ$-set of the Scott space of $Q$. In this paper, we give a negative answer to Martin's long standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for $ω$-algebraic domains.

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arXiv2023

Further studies on open well-filtered spaces

The open well-filtered spaces were introduced by Shen, Xi, Xu and Zhao to answer the problem whether every core-compact well-filtered space is sober. In the current paper we explore further properties of open well-filtered spaces. One of the main results is that if a space is open well-filtered, then so is its upper space (the set of all nonempty saturated compact subsets equipped with the upper Vietoris topology). Some other properties on open well-filtered spaces are also studied.

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arXiv2022

On Scott power spaces

In this paper, we mainly discuss some basic properties of Scott power spaces. For a $T_0$ space $X$, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of $X$ endowed with the Smyth order. It is proved that the Scott power space $Σ\mathsf{K}(X)$ of a well-filtered space $X$ is still well-filtered, and a $T_0$ space $Y$ is well-filtered iff $Σ\mathsf{K}(Y)$ is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$. A sober space is constructed for which its Scott power space is not sober. A few sufficient conditions are given under which a Scott power space is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.

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arXiv2022

Not every countable complete lattice is sober

The study of the sobriety of Scott spaces has got an relative long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Not long after, Isbell gave a complete lattice with non-sober Scott space. Based on Isbell's example, Xu, Xi and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. Let $ΣP$ be the Scott space of poset $P$. In this paper, we first prove that the topology of the product space $ΣP\times ΣQ$ coincides with the Scott topology on the product poset $P\times Q$ if the set $Id(P)$ and $Id(Q)$ of all non-trivial ideals of posets $P$ and $Q$ are both countable. Based on this result, we deduce that a directed complete poset $P$ has a sober Scott space, if $Id(P)$ is countable and the space $ΣP$ is coherent and well-filtered. Thus a complete lattice $L$ with $Id(L)$ countable has a sober Scott space. Making use the obtained results, we then construct a countable complete lattice whose Scott space is non-sober and thus give a negative answer to Jung's problem.

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arXiv2021

The reflectivity of some categories of T0 spaces in domain theory

Keimel and Lawson proposed a set of conditions for proving a category of topological spaces to be reflective in the category of all T0 spaces. These conditions were recently used to prove the reflectivity of the category of all well-filtered spaces. In this paper, we prove that, in certain sense, these conditions are not just sufficient but also necessary for a category of T0 spaces to be reflective. Using this general result, we easily deduce that several categories proposed in domain theory are not reflective, thus answered a few open problems.

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arXiv2020

On open well-filtered spaces

We introduce and study a new class of $T_0$ spaces, called open well-filtered spaces. The main results we proved include (1) every well-filtered space is an open well-filtered space; (2) every core-compact open well-filtered space is sober. As an immediate corollary, we deduce that every core-compact well-filtered space is sober. This provides another different and relatively more straight forward method to answer the open problem posed by Jia and Jung: Is every core-compact well-filtered space sober?

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arXiv2020

$ω$-Rudin spaces, well-filtered determined spaces and first-countable spaces

We investigate some versions of $d$-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a $T_0$ space $X$ is first-countable, then $X$ is an $ω$-Rudin space; (ii) every $ω$-well-filtered space is sober if its sobrification is first-countable; (iii) if a $T_0$ space is second-countable or first-countable and with a countable underlying set, then it is a $ω$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $ω$-well-filtered space is countably-directed; (vi) every first-countable $ω$-well-filtered $ω^\ast$-$d$-space is sober.

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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.

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