Zhao, Dongsheng

A nonempty compact saturated subset F of a topological space is called k-irreducible if it cannot be written as a union of two compact saturated proper subsets. A topological space is said to be co-sober if each of its k-irreducible compact saturated sets is the saturation of a point. Wen and Xu (2018) proved that Isbell's non-sober complete lattice equipped with the lower topology is sober but not co-sober. So far, it is still unknown whether every sober Scott space is co-sober. In this paper, we construct a dcpo whose Scott space is sober but not co-sober, which strengthens Wen and Xu's result.

One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo.

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 summarizing survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces.

A topological space has a domain model if it is homeomorphic to the maximal point space Max(P) of a domain P . Lawson proved that every Polish space X has an ω -domain model P and for such a model P , Max(P) is a Gδ -set of the Scott space of P. Martin (2003) then asked whether it is true that for every ω -domain Q, Max(Q) is Gδ-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 ω -algebraic domains. In addition, we also construct an ω -ideal domain Q˜ for the constructed Q such that their maximal point spaces are homeomorphic. Therefore, Max(Q) is a Gδ -set of the Scott space of the new model Q˜.

A topological space is sober if every nonempty irreducible closed set is the closure of a unique singleton set. Sobriety is precisely the topological property that allows one to recover completely a topological space from its frame of opens. Because every Hausdor space is sober, sobriety is an overt, and hence unnamed, notion. Even in non-Hausdor settings, sober spaces abound. A well-known instance of a sober space appears in domain theory: the Scott topology of a continuous dcpo is sober. The converse is false as witnessed by two counterexamples constructed in the early 1980's: the first by P.T. Johnstone and the second (a complete lattice) by J. Isbell. Since then, there has been limited progress in the quest for an order-theoretic characterization of those dcpo's for which their Scott topology is sober. This paper provides one answer to this open problem.

Based on the topological Rudin's Lemma, we introduce the notions of Rudin set and well-filtered determined set in a topological space. Using such sets, we formulate and prove some new characterizations of well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on well-filtered spaces. Our study also lead to the definition of a new class of spaces - the strong d-spaces, and some problems whose solutions will strengthen the understanding of the related structures.

A poset model of a topological space X is a poset P such that X is homeomorphic to the maximal point space of P (the set Max(P) of all maximal points of P equipped with the relative Scott topology of P). The poset models of topological spaces based on other topologies, such as Lawson topology and lower topology, have also been investigated by other people. These models establish various types of new links between posets and topological spaces. In this paper we introduce the strong Scott topology on a poset and use it to de ne the strong poset model: a strong poset model of a space X is a poset P such that Max(P) (equipped with the relative strong Scott topology) is homeomorphic to X. The main aim is to establish a characterization of T1 spaces with T-generated topologies (such as the Hausdor k-spaces) in terms of maximal point spaces of posets. A poset P is called ME-separated if for any elements x; y of P, x y i " y \ Max(P) "x \ Max(P). We consider the topological spaces that have an ME-separated strong poset model. The main result is that a T1 space has an ME-separated strong poset model i its topology is T-generated. The class of spaces whose topologies are T-generated include all Scott spaces and all Hausdor k-spaces.

A dcpo model of a topological space X is a dcpo (directed complete poset) P such that X is homeomorphic to the maximal point space of P with the subspace topology of the Scott space of P. It has been proved previously by X. Xi and D. Zhao that every T1 space has a dcpo model. It is, however, still unknown whether every T1 space has a bounded complete dcpo model (a poset is bounded complete if each of its upper bounded subsets has a supremum). In this paper we rst show that the set of natural numbers equipped with the co- nite topology does not have a bounded complete dcpo model, then prove that a large class of topological spaces (including all Hausdorff k-spaces) have a bounded complete dcpo model. We shall mainly focus on the model formed by all the nonempty closed compact subsets of the given space.

We prove that for every T0 space X, there is a well-filtered space W(X) and a continuous mapping ηx : X⭢ W(X), such that for any well-filtered space Y and any continuous mapping 𝒇 : X⭢Y there is a unique continuous mapping 𝒇^: W(X)⭢Y such that 𝒇=𝒇^∘ηX. Such a space W(X) will be called the well-filterification of X. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.