Related papers: 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…

In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress. In this paper, we shall present a brief summarising survey on some of such development. Furthermore, we shall pose and…

We first introduce and study two new classes of subsets in $T_0$ spaces - $\omega$-Rudin sets and $\omega$-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed…

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…

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…

Based on topological Rudin's Lemma, we investigate two new kinds of sets - Rudin sets and well-filtered determined sets in $T_0$ topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces…

We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $\eta_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous…

We prove that a $T_0$ topological space is $\omega$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this,…

The Hofmann-Mislove theorem states that in a sober space, the nonempty Scott open filters of its open set lattice correspond bijectively to its compacts saturated sets. In this paper, the concept of $c$-well-filtered spaces is introduced.…

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…

M. Escard\'o et al. asked whether the core compactly generated topology of a sober space is again sober and the sobrification of a core compactly generated space again core compactly generated. In this note, we answer the problem by…

In this paper, we obtain some sufficient conditions for the D-completion of a T0 space to be the well-filterification of this space, the well-filterification of a T0 space to be the sobrification of this space and the D-completion of a T0…

In this paper, we investigate the sobriety of weakly first-countable spaces and give some sufficient conditions that the Scott topologies of the open set lattices are sober. The main results are: (1) Let $P$ and $Q$ be two posets. If…

Recently, Xu proposed a strongly well-filtered space in [24] and systematically investigated some of its properties and characterizations. In this paper, we introduce a new class of T0-spaces called S*-well-filtered spaces, which is…

In this paper, we provide a uniform approach to $d$-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. For a subset system H, the theory of H-sober spaces and super H-sober…

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…

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…

Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of $T_0$ spaces instead of restricting to posets. In this paper, we respond to this calling by…

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…

K\"unzi and Ferrario have shown that a $T_0$ space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictly zero-dimensional biframe is a congruence biframe if…