Zhao, Dongsheng

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.

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.

A dcpo P is continuous if and only if the lattice C(P) of all Scott-closed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C(P). In this paper, we study the order-theoretic properties of C(P) for general dcpo's P. The main results are: (i) every C(P) is C-continuous; (ii) a complete lattice L is isomorphic to C(P) for a complete semilattice P if and only if L is weak-stably C-algebraic; (iii) for any two complete semilattices P and Q, P and Q are isomorphic if and only if C(P) and C(Q) are isomorphic. In addition, we extend the function P 7! C(P) to a left adjoint functor from the category DCPO of dcpo's to the category CPAlg of C-prealgebraic lattices.

Posing good problems is important for learning, teaching and research in mathematics. In this paper, the converse problem posing strategy is applied to Gauss's method that has been used to obtain the summation formula of an Arithmetic Progression. The work here serves as a simple but typical example to demonstrate the use of this strategy. The results obtained may also help the reader see to what extent Gauss's method can be applied, thus enriching one's understanding of this famous method.

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.

We prove that (1) for any complete lattice 𝓛, the set 𝒟 (𝓛) of all non-empty saturated compact subsets of the Scott space of 𝓛 is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice 𝓛 is non-sober, then the Scott space of 𝒟 (𝓛) is non-sober. Using these results and Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus giving an affirmative answer to a problem posed by Achim Jung. We also prove that 𝚊 𝒯₀ space is well-filtered iff its upper space (the set 𝒟 (𝓧) of all non-empty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space P is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.