Abstract A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ . Lawson proved that every Polish space $X$ has an $\omega$ -domain model $P$ and for such a model $P$ , $\mbox{Max}(P)$ is a $G_{\delta }$ -set of the Scott space of $P$ . Martin (2003) then asked whether it is true that for every $\omega$ -domain $Q$ , $\mbox{Max}(Q)$ is $G_{\delta }$ -set of the Scott space of $Q$ . In this paper, we give a negative answer to Martin’s long-standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for $\omega$ -algebraic domains. In addition, we also construct an $\omega$ -ideal domain $\widetilde{Q}$ for the constructed $Q$ such that their maximal point spaces are homeomorphic. Therefore, $\textrm{Max}(Q)$ is a $G_\delta$ -set of the Scott space of the new model $\widetilde{Q}$ .

Abstract In this paper, we introduce a new class of $T_0$ spaces called wb-sober spaces, which is strictly larger than the class of open well-filtered spaces. Unlike open well-filtered spaces, wb-sober spaces are defined more intuitively by requiring certain special subsets, termed wb-irreducible closed sets, to have singleton closures. We establish several key results about these spaces, including (1) every open well-filtered space is wb-sober, but not vice versa; (2) every strongly core-coherent wb-sober space is open well-filtered; (3) a space is core-compact iff its irreducible closed sets are wb-irreducible, providing a characterization of core-compactness; (4) every core-compact wb-sober space is sober, thereby generalizing the Jia-Jung problem. In addition, we investigate the core-coherence of the Xi-Zhao model. We prove that a $T_1$ space contains finite number of isolated points iff its Xi-Zhao model is core-coherent iff its Xi-Zhao model is strongly core-coherent. Based on this result, we then propose a general approach to constructing a non-routine open well-filtered but not well-filtered dcpo.

Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered.

Pointwise pseudoquasi-metrics play an important role in the theory of lattice-valued topology ($L$-topology). Bearing in mind that closed balls and their relations with pseudoquasi-metrics have historically attracted the attention of mathematicians, it is very surprising that no attention has been paid to the relations between pointwise pseudoquasi-metrics and closed balls. In this article, we first introduce the concept of pointwise closed-ball systems and prove that the resulting category is isomorphic to that of pointwise pseudoquasi-metrics. Subsequently, we study the topological properties of pointwise pseudoquasi-metrics via pointwise closed-ball systems. Interestingly, the$L$-topologies defined by open sets and complements of closed sets coincide for any pointwise pseudometric. Finally, we expose an important theoretical application of the pointwise closed-systems in providing a different and relatively simpler proof of the celebrated metrization theorem of the$L$-fuzzy real line.