arXiv subjects

Weng Kin Ho

Publications and source records attributed to Weng Kin Ho.

10 recordsLinked to original sources

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 $\alpha$ 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↗

arXiv2019

Topological Scott Convergence Theorem

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 with an attempt to formulate a topological version of the Scott Convergence Theorem, i.e., an order-theoretic characterisation of those posets for which the Scott-convergence $\mathcal{S}$ is topological. To do this, we make use of the $\mathcal{ID}$ replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., $\mathcal{I}$-continuous spaces correspond to continuous posets, as $\mathcal{I}$-convergence corresponds to $\mathcal{S}$-convergence. In this paper, we consider two novel topological concepts, namely, the $\mathcal{I}$-stable spaces and the $\mathcal{DI}$ spaces, and as a result we obtain some necessary (respectively, sufficient) conditions under which the convergence structure $\mathcal{I}$ is topological.

cs.LO↗

arXiv2018

Domains via approximation operators

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

cs.LO↗

arXiv2018

Generalised Net Convergence Structures in Posets

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 a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ is $\mathcal{R}^*$-doubly continuous.

math.GN↗

arXiv2018

The Ho-Zhao Problem

Given a poset $P$, the set, $Γ(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $Γ$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $Γ(P) \cong Γ(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $Γ$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $Γ$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $Γ$-faithful. This subcategory subsumes all currently known $Γ$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

cs.LO↗

arXiv2017

On A New Convergence Class in Sup-sober Spaces

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 proving a topological parallel of a 2005 result due to B. Zhao and D. Zhao, i.e., an order-theoretic characterisation of those posets for which the Scott-convergence is topological. We do this by adopting a recent approach due to D. Zhao and W. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class $\mathcal{I}$ in $T_0$-spaces called ${\operatorname{Irr}}$-convergence and establish that a sup-sober space $X$ is ${\operatorname{SI}}^{-}$-continuous if and only if it satisfies $*$-property and the convergence class $\mathcal{I}$ in it is topological.

cs.LO↗

arXiv2016

Strong completions of spaces

A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respect to the specialization order) of a $T_0$ space are always irreducible, and (4) the topological closure of every irreducible set is again irreducible. In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2009) developed the core of domain theory directly in the context of $T_0$ spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo -- a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho (2009). The topological counterpart of a dcpo is thus this: A $T_0$ space is said to be strongly complete if the suprema of all irreducible subsets exist. In this paper, we show that the category, $\mathbf{scTop^+}$, of strongly complete $T_0$ spaces forms are reflective subcategory of a certain lluf subcategory, $\mathbf{Top^+}$, of $T_0$ spaces.

cs.LO↗

arXiv2016

Join-continuity + Hypercontinuity = Prime continuity

A remarkable result due to Kou, Liu & Luo states that the condition of continuity for a dcpo can be split into quasi-continuity and meet-continuity. Their argument contained a gap, however, which is probably why the authors of the monograph Continuous Lattices and Domains used a different (and fairly sophisticated) sequence of lemmas in order to establish the result. In this note we show that by considering the Stone dual, that is, the lattice of Scott-open subsets, a straightforward proof may be given. We do this by showing that a complete lattice is prime-continuous if and only if it is join-continuous and hypercontinuous. A pleasant side effect of this approach is that the characterisation of continuity by Kou, Liu & Luo also holds for posets, not just dcpos.

cs.LO↗

arXiv2016

On a new convergence class in k-bounded sober spaces

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 proving a topological parallel of a 2005 result due to B. Zhao and D. Zhao, i.e., an order-theoretic characterisation of those posets for which the lim-inf convergence is topological. We do this by adopting a recent approach due to D. Zhao and W. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class on $T_0$ spaces called Irr-convergence and established that this convergence class $\mathcal{I}$ on a $k$-bounded sober space $X$ is topological if and only if $X$ is Irr-continuous.

cs.LO↗