arXiv subjects

Xinpeng Wen

Publications and source records attributed to Xinpeng Wen.

7 recordsLinked to original sources

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

On GSI2-convergence in T0-spaces

In this paper,we introduce the concept of GSI$_2$-convergence in $T_0$ spaces and the related concept of (strongly) QI$_2$-continuous spaces. It is proved that if GSI$_2$-convergence in $X$ is topological iff $X$ is strongly QI$_2$-continuous for any irreducible complete $T_0$ space $X$.

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

Non-reflective categories of some kinds of weakly sober spaces

Erné weakened the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced and discussed three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. Three other kinds of non-sober spaces, namely $\mathsf{DC}$ space, $\mathsf{RD}$ space and $\mathsf{WD}$ space, were introduced and investigated by Xu, Shen, Xi and Zhao. All these six kinds of spaces are strictly weaker than sober spaces. In this paper, it is shown that none of the category of all $\mathsf{DC}$ spaces, that of all $\mathsf{RD}$ spaces, that of all $\mathsf{WD}$ spaces, that of all quasisober spaces, that of all weakly spaces and that of all cut spaces is reflective in the category of all $T_0$ spaces with continuous mappings.

math.GM↗

arXiv2019

On some kinds of weakly sober spaces

In \cite{E_2018}, Erné relaxed the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. In this paper, their basic properties are investigated. It is shown that some properties which are similar to that of sober spaces hold and others do not hold.

math.GN↗