Zhao, Dongsheng

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˜.

Posing or raising appropriate problems is necessary and important for active and deep learning in mathematics. However, students rarely make effort to find thinking problems by themselves. Such an attitude and behavior often lead to passive learning and cause various difficulties and problems in mathematics teaching. The main objective of this paper is to explore the ways to develop students’ ability to find and pose good mathematical problems and thus to promote more active learning in mathematics.

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.

The Singapore Mathematics Assessment and Pedagogy Project (SMAPP) is a research project conducted by the National Institute of Education and funded by the Ministry of Education. It aims to make assessment practices an integral part of teaching and learning, and broaden student learning outcomes by using authentic disciplinary tasks. As part of the project, some guidelines are provided for designing disciplinary tasks which have the distinctive features of their emphasis on contextual aspects. One of the criteria of a good disciplinary task is its ability to assess multiple mathematical competencies of students. In this paper, we will present some examples to illustrate how these competencies can be assessed. Another aim is to find out to what extent these tasks serve the purpose of assessing these competencies, by analyzing the students’ performance in a sample SMAPP task.

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.

In this paper we investigate -primary ideals which unify prime ideals and primary ideals. A number of main results about prime ideals and primary ideals are extended into this general framework. Prime ideals and primary ideals are two of the most important structures in commutative algebra. Although different from each other in many aspects, they share quite a number of similar properties as well( see [1] ). However, these two structures have been treated rather differently, and all of their properties were proved separately. It is therefore natural to examine whether it is possible to have a unified approach to studying these two structures. In this short paper we introduce the notion of -primary ideals where is a mapping that assigns to each ideal I an ideal (I) of the same ring. Such -primary ideals unify the prime and primary ideals under one frame. This approach clearly reveals how similar the two structures are and how they are related to each other. In the first section, we introduce ideal expansion and define primary ideals with respect to such an expansion. Besides the familiar expansions 0, 1 and B, we also have a new expansion M defined by means of maximal ideals. In the second section, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized primary ideals with respect to such expansions. In this paper, all the rings used are commutative rings with an multiplication identity and all the ring homomorphisms preserve the identity. We shall use Id(R) to denote the set of all ideals of the ring R.

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.

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.

In this paper, we investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. It is proved that every space with a first-countable sobrification is an ω-Rudin space and every first-countable space is well-filtered determined. Therefore, every ω-well-filtered space with a first-countable sobrification is sober. It is also shown that every irreducible closed subset in a first-countable ω-well-filtered space is countably directed, hence every first-countable ω*-well-filtered d-space is sober.