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Dongsheng Zhao

Publications and source records attributed to Dongsheng Zhao.

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.

math.GN↗
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 $P$.

math.GN↗
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.

math.GN↗
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.

math.GN↗
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.

math.GN↗
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.

math.GN↗
arXiv2025

Sober topologies on a set

The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff topology. This property plays a crucial role in characterizing the spectral spaces of commutative rings and topological spaces determined by their lattices of open sets. In this paper, we investigate the statute of sober topologies in the complete lattice of all topologies on a given set. The main results to be proved include: (1) every T1 topology is the join of some sober topologies; (2) every topology is the meet of some sober topologies; (3) the set of all sober topologies is directed complete; (4) every Alexanderoff - discrete topology is the meet of some sober Alexanderoff - discrete topologies; (5) the minimal sober topologies are exactly the Scott topologies of sup-complete chains; (6) an example will be constructed to show that the intersection of a decreasing sequence of Hausdorff topologies need not be sober.

math.GN↗
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.

math.GN↗
arXiv2024

Extraction of Weak Surface Diaphragmatic Electromyogram Using Modified Progressive FastICA Peel-Off

Diaphragmatic electromyogram (EMGdi) contains crucial information about human respiration therefore can be used to monitor respiratory condition. Although it is practical to record EMGdi noninvasively and conveniently by placing surface electrodes over chest skin, extraction of such weak surface EMGdi (sEMGdi) from great noisy environment is a challenging task, limiting its clinical use compared with esophageal EMGdi. In this paper, a novel method is presented for extracting weak sEMGdi signal from high-noise environment based on fast independent component analysis (FastICA), constrained FastICA and a peel-off strategy. It is truly a modified version of of progressive FastICA peel-off (PFP) framework, where the constrained FastICA helps to extract and refine respiration-related sEMGdi signals, while the peel-off strategy ensures the complete extraction of weaker sEMGdi components. The method was validated using both synthetic and clinical signals. It was demonstrated that our method was able to extract clean sEMGdi signals efficiently with little distortion. It outperformed state-of-the-art comparison methods in terms of sufficiently high SIR and CORR at all noise levels when tested on synthetic data, while also achieved an accuracy of 95.06% and a F2-score of 96.73% for breath identification on clinical data. The study presents a valuable solution for noninvasive extraction of sEMGdi signals, providing a convenient and valuable way of ventilator synchrony with a significant potential in aiding respiratory rehabilitation and health.

physics.med-ph↗
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.

math.GN↗
arXiv2023

The sheaf representation of residuated lattices

The residuated lattices form one of the most important algebras of fuzzy logics and have been heavily studied by people from various different points of view. Sheaf presentations provide a topological approach to many algebraic structures. In this paper, we study the topological properties of prime spectrum of residuated lattices, and then construct a sheaf space to obtain a sheaf representation for each residuated lattice.

math.GN↗
arXiv2023

One-step closure, weak one-step closure and meet continuity

This paper studies the weak one-step closure and one-step closure properties concerning the structure of Scott closures. We deduce that every quasicontinuous domain has weak one-step closure and show that a quasicontinuous poset need not have weak one-step closure. We also constructed a non-continuous poset with one-step closure, which gives a negative answer to an open problem posed by Zou et al.. Finally, we investigate the relationship between weak one-step closure property and one-step closure property and prove that a poset has one-step closure if and only if it is meet continuous and has weak one-step closure.

math.GN↗
arXiv2023

Quasiexact posets and the moderate meet-continuity

The study of weak domains and quasicontinuous domains leads to the consideration of two types generalizations of domains. In the current paper, we define the weak way-below relation between two nonempty subsets of a poset and quasiexact posets. We prove some connections among quasiexact posets, quasicontinuous domains and weak domains. Furthermore, we introduce the weak way-below finitely determined topology and study its links to Scott topology and the weak way-below topology first considered by Mushburn. It is also proved that a dcpo is a domain if it is quasiexact and moderately meet continuous with the weak way-below relation weakly increasing.

math.GN↗
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.

math.GN↗
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.

math.GN↗
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.

math.GN↗