Related papers: On sobriety of Scott topology on dcpos
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…
By Thron, a topological space $X$ has the property that $C(X)$ isomorphic to $C(Y)$ implies $X$ is homeomorphic to $Y$ iff $X$ is sober and $T_D$, where $C(X)$ and $C(Y)$ denote the lattices of closed sets of $X$ and $T_0$ space $Y$,…
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…
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…
In analogy to a result due to Drake and Thron about topological spaces, this paper studies the dcpos (directed complete posets) which are fully determined, among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will be…
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…
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…
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…
Inspired by Zhao and Xu's study on which a dcpo can be determined by its Scott closed subsets lattice, we further investigate whether a poset (or dcpo) $P$ is able to be determined by the family $\mathcal Q(P)$ of its Scott compact…
We construct a complete lattice $Z$ such that the binary supremum function $\sup:Z\times Z\to Z$ is discontinuous with respect to the product topology on $Z\times Z$ of the Scott topologies on each copy of $Z$. In addition, we show that…
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.…
We define a class of subsets of a topological space that coincides with the class of compact saturated subsets when the space is sober, and with enough good properties when the space is not sober. This class is introduced especially in view…
Let $Y$ and $Z$ be two given topological spaces, ${\cal O}(Y)$ (respectively, ${\cal O}(Z)$) the set of all open subsets of $Y$ (respectively, $Z$), and $C(Y,Z)$ the set of all continuous maps from $Y$ to $Z$. We study Scott type topologies…
We investigate when the space $\mathcal O_X$ of open subsets of a topological space $X$ endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of $X$, which in turn is characterized in…
In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give…
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of…
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of…
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…
The notion of sobriety is extended to the realm of topological spaces valued in a commutative and unital quantale, via an adjunction between a category of quantale modules and the category of quantale-valued topological spaces. Relations…
We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive…