Ho, Weng Kin

A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respect to the specialization order) of a T0 space are always irreducible, and (4) the topological closure of every irreducible set is again irreducible. In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2009) developed the core of domain theory directly in the context of T0 spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo – a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho. The topological counterpart of a dcpo is thus this: A T0 space is said to be strongly complete if the suprema of all irreducible subsets exist. In this paper, we show that the category, scTop+, of strongly complete To spaces forms a reflective subcategory of a certain lluf subcategory, Top+, of To spaces.

Given a poset P, the set, Γ(P) of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory C of Posd (the category of posets and Scott-continuous maps) is said to be Γ-faithful if for any posets P and Q in C, Γ(P)≅Γ(Q) implies P≅Q. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are Γ-faithful, while Posd is not. Ho & Zhao (2009) asked whether the category DCPO of dcpos is Γ-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is Γ-faithful. This subcategory subsumes all currently known Γ-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

Computational Thinking is a paradigm for problem solving with the goal that problems and their solutions can be executed by a computer. Because of one’s natural association of computer and computer programming, one is often misguided to think that computational thinking is solely reserved for the computer scientists and computer programmers. This chapter takes the stance that computational thinking is a generically useful way of thinking that is applicable across all disciplines, and in particular, mathematics. We highlight four design principles that mathematics teachers in Secondary Schools and Junior Colleges can apply to create lessons that promote computational thinking to forge mathematical ideas and enhance mathematics learning, which we term as “Math + C” lessons.

This article focuses on a challenging geometry problem that was originally posed to primary school students. Four solution approaches, ranging from elementary to advanced, are discussed. Reflections on these approaches and the problem solving processes are also shared.

A number of authors have exported domain-theoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of non-trivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features.

This position paper claims that the way a mathematical problem is solved depends on the technology available to the problem-solver then. Drawing on the authors’ mathematical experience of finding a new solution to an old problem - the Generalised Fermat Point Problem, salient observations are drawn to illustrate how a problem solver’s experience can shaped by technological affordances.

Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of T0 spaces instead of restricting to posets. In this paper, we respond to this calling with an attempt to formulate a topological version of the Scott Convergence Theorem, i.e., an order-theoretic characterisation of those posets for which the Scott-convergence S is topological. To do this, we make use of the ID replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., I-continuous spaces correspond to continuous posets, as I-convergence corresponds to S-convergence. In this paper, we consider two novel topological concepts, namely, the I-stable spaces and the DI spaces, and as a result we obtain some necessary (respectively, sufficient) conditions under which the convergence structure I is topological.

This tutorial-style paper aims to give a quick beginner’s introduction to the theory of continuous valuations – a domain-theoretic variant of measure theory tailored for topological spaces and ordered structures. Central to the theory of continuous valuations is the famous theorem of Claire Jones that asserts that the probabilistic powerdomain (i.e., the space of continuous valuations arising from the Scott topology of domains) ordered by the stochastic ordering is again a domain. Most existing proofs of Jones’ theorem are fairly technical and involved, and thus this paper aims to present an intuitive and self-contained proof that allows reader quick access to it with minimum mathematical overhead. Sifting through all the concepts and theorems leading up to Jones’ theorem, we make a judicious choice of a core essence – constituting those results that are featured prominently in the theory of continuous valuations – and expound on their proofs, focusing on the key techniques and reasoning methods while contracting on, or even omitting, other peripheral or burdensome intricacies.

The Concrete-Pictorial-Abstract (CPA) approach, based on Bruner’s conception of the enactive, iconic and symbolic modes of representation, is a well-known instructional heuristic advocated by the Singapore Ministry of Education since early 1980’s. Despite its ubiquity as a teaching strategy throughout the entire mathematics education community in Singapore, it is somewhat surprising to see a lack of an account of its theoretical roots. This paper is an attempt to contribute to this discussion on the CPA strategy and its potential role in continuing advancement of quality mathematics education.

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.