Zhao, Dongsheng

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.

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 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.

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.

The main objective of this project is to explore the various applications of problem posing strategy in promoting active learning in higher mathematics, especially in modern algebra and commutative algebra.

We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_{0}$ space $(X,\tau)$ is a quasicontinuous space if and only if $SI(X)$ is locally hypercompact if and only if $(\tau_{SI},\subseteq)$ is a hypercontinuous lattice; (2) a $T_{0}$ space $X$ is an $SI$-continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$-space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous space if and only if it is a quasicontinuous domain under the specialization order; (4) there exists an adjunction between the category of quasicontinuous domains and the category of quasicontinuous spaces which are well-filtered posets under their specialization orders.

In domain theory, by a poset model of a T1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ω(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P\↑x, x ∈ P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Maxω(P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T1 spaces and order structures. The main results proved in this paper include (i) a T1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is second-countable if and only if it has an ω-algebraic poset LT-model; (iii) every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets.