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Xiaoquan Xu

Publications and source records attributed to Xiaoquan Xu.

At least 19 recordsLinked to original sources
arXiv2026

Scott topologies on meet-continuous domains

We study Scott products and sobriety of countable meet-continuous domains, meaning meet-continuous dcpos without any additional continuity or least-element assumption. Using the complete-lattice test-family theorem of Xu and Ji, we prove finite-product equality for those domains that are $L$-dcpos, and for the weaker class whose principal ideals have suprema of all nonempty subsets. For an arbitrary family of nonempty countable meet-continuous $L$-dcpos, we prove that the Scott topology on the order product equals the product of the factor Scott topologies if and only if only finitely many factors lack a least element. We also establish sobriety under bounded completeness and under additional common-upper-bound conditions. Assuming square-product equality, sobriety is characterized by Scott closedness of common-upper-bound sections associated with irreducible Scott-closed sets, with an equivalent sequential formulation in the countable case. Two extraction lemmas extend to meet-semilattice dcpos. The finite-product and sobriety questions for general countable meet-continuous domains remain unresolved here.

math.LO↗
arXiv2026

Every countable meet-continuous lattice is Scott sober

We prove that every countable meet-continuous lattice is Scott sober. For any family of such lattices, the Scott topology on the product equals the product of the factor Scott topologies, and the product Scott space is sober. These results answer affirmatively Questions 7.1 and 7.2 of Xu (arXiv:2609.18032v1), extending the countable-frame results beyond finite distributivity. A diagonal argument on meet coordinates extracts directed punctured intervals from nonprincipal ideals. Join translates of these intervals form a countable family detecting Scott openness and yield Scott open rectangles by successive finite choices. We also show that the cardinal spectrum of Scott non-sober meet-continuous lattices is upward closed and, under the Continuum Hypothesis, consists of all uncountable cardinals.

math.LO↗
arXiv2026

Scott topologies on products of countable complete Heyting algebras

We prove that the Scott topology commutes with arbitrary products of countably presented frames and that every such product has a sober Scott space. In particular, this holds for arbitrary products of countable complete Heyting algebras. For a family of consonant spaces, Scott-product compatibility of their open-set lattices is equivalent to consonance of their topological sum. Countable generation does not suffice: a countably generated spatial frame can have a non-sober Scott space and fail the product identity for its square. We also show that the cardinal spectra of Scott non-sober frames and spatial frames are upward closed and, under the Continuum Hypothesis, consist of all uncountable cardinals. Finally, every Artinian $T_0$ web space is a $B$-space; if it is also a $d$-space, it carries the Scott topology of an algebraic dcpo. Consequently, every Artinian meet-continuous dcpo is algebraic. This yields a dichotomy theorem: a dcpo with a non-sober Scott space must fail meet continuity or Artinianity.

math.LO↗
arXiv2026

School of Mathematics and Statistics, Guilin University of Technology, Guilin 541004, ChinaScott spaces of complete Boolean algebras need not be co-sober

In this paper, we first prove that for a complete Boolean algebra $L$, the complement graph of $L$ is a \emph{KC}-space (as a subspace) and each non-singleton compact irreducible subspace of that graph generates a non-principal $k$-irreducible compact saturated set in the Scott space of the square algebra $L\times L$. We then show that every compact sequential \emph{US}-space embeds into the complement graph of a suitable complete Boolean algebra. The embedding is built from a finite tail-constraint poset and its regular-open completion. Applying the construction to van Douwen's compact Fréchet anti-Hausdorff \emph{US}-space gives a complete Boolean algebra $B$ whose Scott space $Σ~\!\!B$ is not co-sober, thereby answering negatively a question on Scott spaces of complete Boolean algebras. The same Scott space $Σ~\!\!B$ is non-sober and, as a Scott space of a complete lattice, is well-filtered.

math.GN↗
arXiv2026

Bona: Automatic Management of Dirty Ancilla Borrowing in Quantum Circuits

The management of ancilla qubits has become a critical technique for reducing quantum circuit width. Dirty ancillas, which may be borrowed from any temporarily idle qubit regardless of their initial states, offer substantial flexibility for width optimization, but their use has so far required manual and error-prone handling. We formalize the dirty-qubit borrowing problem and establish a fundamental computational limit by proving its NP-hardness. To support practical optimization, we present \bona, the first scheduler for dirty-qubit borrowing, built on a novel depth-aware heuristic algorithm. We evaluate \bona~ across a variety of benchmarks, including practical quantum circuits and randomly arranged compositions of real circuit modules, and find that it reduces nearly 99\% of dirty ancillas on average with controlled depth overhead. In particular, for parallel quantum walk---an essential component of parallel Hamiltonian simulation---\bona~ matches the circuit width achieved by the clean-qubit schemes of \citeauthor{jiang2024recycling}~(\citeyear{jiang2024recycling}) and \citeauthor{quantinuum}~(\citeyear{quantinuum}), but attains significantly smaller circuit depth, providing concrete evidence that dirty ancillas offer unique optimization advantages in circuits with certain parallelism.

cs.PL↗
arXiv2026

Characterization of $T_0$-spaces for quasi-liminf convergence being topological

The authors' primary goal in this paper is to extend some important results related to the liminf-convergence and $\mathcal{QS}$-convergence in domain theory to the setting of $T_0$-spaces. To that end, we study the quasi-liminf convergence in $T_0$-spaces and introduce a new kind of $T_0$-spaces --- weakly locally hypercompact spaces (shortly \emph{WLH}-spaces). It is proved that every locally hypercompact $T_0$-space is a \emph{WLH}-space, and a $T_0$-space $(X, τ)$ is a \emph{WLH}-space iff the quasi-liminf convergence in $(X, τ)$ is topological. Hence the quasi-liminf convergence in a locally hypercompact space is topological, and for a quasicontinuous poset $P$, the quasi-liminf convergence is topological and agrees with convergence in the Lawson topology $λ(P)$. We also show that a $T_0$-space $(X,τ)$ is locally hypercompact iff the $\mathcal{QS}$-convergence in $(X,τ)$ coincides with the convergence in the topology $τ$. Therefore, a poset $P$ is quasicontinuous iff $\mathcal{QS}$-convergence in the Scott space of $P$ is topological iff $\mathcal{QS}$-convergence coincides with convergence in the Scott topology $σ(P)$. Using the quasi-liminf convergence, we give several characterizations of $C$-spaces and continuous posets.

math.GN↗
arXiv2026

On strong $R$-spaces

In this paper, we mainly investigate some basic properties of strong $R$-spaces. It is shown that the property of being a strong $R$-space is closed-hereditary, saturated-hereditary and retractive, but not finite productive. Hence the category $\mathbf{S}$-$\mathbf{Top}_r$ of strong $R$-spaces and continuous mappings is not reflective in the category $\mathbf{Top}_0$ of $T_0$-spaces and continuous mappings. It is proved that a $T_0$-space $(X, τ)$ is a strong $R$-space iff every nonempty $τ$-closed subset of $X$ is compact in $(X, τ^{d})$, where $τ^d$ is the de Groot dual of $τ$; consequently, if $(X, τ)$ is a strong $R$-space (especially, if $(X, τ)$ is a coherent well-filtered space), then $τ\subseteq τ^{dd}$. Therefore, for any locally compact strong $R$-space $(X, τ)$, we have $τ=τ^{dd}$. Finally, we investigate conditions under which the Smyth power space and Scott power space of a $T_0$-space is a strong $R$-space. Several such conditions are given.

math.GN↗
arXiv2026

On three problems about well-filteredness of $T_0$-spaces

In this paper, we show that there is a countable Noetherian complete lattice $L$ and an order-compatible $d$-topology $τ$ on $L$ such that $(L, τ)$ is not well-filtered, and there exist a dcpo $P$ and an order-compatible well-filtered topology $τ$ on $P$ but the Scott topology $σ(P)$ is not well-filtered. For such poset $P$ and topology $τ$, let $Y=(P, τ)$ and $X = 1$ (the topological space with single point), then the function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is not well-filtered. These results answer three open problems concerning the well-filteredness of $T_0$-spaces.

math.GN↗
arXiv2026

Some new results on well-filteredness of $T_0$-spaces

For a $T_0$-space $X$, let $Q (X)$ be the poset of nonempty compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if it satisfies the following two conditions: (1) $\wedge K$ exists for any $K\in Q(X)$, and (2) for any filtered family $\{K_d : d\in D\}\subseteq Q(X)$ and $x\in X$, if $\bigvee^{\uparrow}_{d\in D}\bigwedge K_d$ exists and $x\not\leq \bigvee^{\uparrow}_{d\in D}\bigwedge K_d$, then there is $φ\in \prod\limits_{d\in D}\!\!K_d$ and an upper bound $u$ of $φ(D)$ such that $x\not\leq u$. In this paper, we prove that every $d$-space with property Q is well-filtered and the Smyth power space of a $T_0$-space always has property Q. Hence the Smyth power construction preserves the well-filteredness. For a complete lattice $L$ and an order-compatible $d$-topology $τ$ on it, we show that when $L$ possesses a certain distributivity, $(L, τ)$ is well-filtered.

math.GN↗
arXiv2025

Strong well-filteredness of upper topology on sup-complete posets

We first introduce and investigate a new class of $T_0$ spaces -- strong R-spaces, which are stronger than both R-spaces and strongly well-filtered spaces. It is proved that any sup-complete poset equipped with the upper topology is a strong R-space and the Hoare power space of a $T_0$-space is a strong R-space. Hence the upper topology on a sup-complete poset is strongly well-filtered and the Hoare power space of a $T_0$-space is strongly well-filtered, which answers two problems recently posed by Xu.

math.GN↗
arXiv2025

Countable quasicontinuous domains are quasialgebraic

We prove that every quasicontinuous domain that fails to be quasialgebraic admits the unit interval [0, 1] as its monotone Lawson-continuous image. As a result, every countable quasicontinuous domain is quasialgebraic.

math.GN↗
arXiv2023

$SI_2$-quasicontinuous spaces

In this paper, as a common generalization of $SI_{2}$-continuous spaces and $s_{2}$-quasicontinuous posets, we introduce the concepts of $SI_{2}$-quasicontinuous spaces and $\mathcal{GD}$-convergence of nets for arbitrary topological spaces by the cuts. Some characterizations of $SI_{2}$-quasicontinuity of spaces are given. The main results are: (1) a space is $SI_{2}$-quasicontinuous if and only if its weakly irreducible topology is hypercontinuous under inclusion order; (2) A $T_{0}$ space $X$ is $SI_{2}$-quasicontinuous if and only if the $\mathcal{GD}$-convergence in $X$ is topological.

math.GN↗
arXiv2022

On sobriety of Scott topology on dcpos

In this paper, we mainly investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Some such conditions are given.

math.GN↗
arXiv2022

A supplement on feathered gyrogroups

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an $ω^ω$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $ω^ω$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, we show that if $G$ has countable $cs^{*}$-character, then it is metrizable; and it is also shown that $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.

math.GN↗
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.

math.GN↗
arXiv2022

On some kinds of factorizable topological groups

Based on the concepts of $\mathbb{R}$-factorizable topological groups and $\mathcal{M}$-factorizable topological groups, we introduce four classes of factorizabilities on topological groups, named $P\mathcal{M}$-factorizabilities, $Pm$-factorizabilities, $S\mathcal{M}$-factorizabilities and $PS\mathcal{M}$-factorizabilities, respectively. Some properties of the four classes of spaces are investigated.

math.GN↗
arXiv2022

On $\mathbf{K}$-reflections of Scott spaces

In this paper, for a full subcategory $\mathbf{K}$ of the category of all $T_0$ spaces with continuous mappings, we investigate the questions under what conditions the $\mathbf{K}$-reflection of a Scott space is still a Scott space and under what conditions the Scott $\mathbf{K}$-completion of a poset exists. Some necessary and sufficient conditions for the $\mathbf{K}$-reflection of a Scott space to be a Scott space and for the existence of Scott $\mathbf{K}$-completion of a poset are established, respectively. It is shown that neither the sobrification nor the well-filtered reflection of the Johnstone space is a Scott space. The $\mathbf{K}$-reflections of Alexandroff spaces and the $\mathbf{K}$-completions of posets are also discussed.

math.GM↗