Now showing 1 - 10 of 29
  • Publication
    Open Access
      374  193
  • Publication
    Open Access
    On topological spaces that have a bounded complete DCPO model
    (2018) ;
    Xi, Xiaoyong
    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.
      108  101
  • Publication
    Open Access
    When exactly is Scott sober?
    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.
      229  211
  • Publication
    Open Access
    Assessing mathematical competencies using disciplinary tasks
    (2012)
    Cheang, Wai Kwong
    ;
    ;
    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.
      220  301
  • Publication
    Open Access
      124  6777
  • Publication
    Open Access
    Learning mathematics through exploration and connection
    (2001) ; ;
    Cheang, Gerald
    ;
    Phang, Rosalind Lay Ping
    ;
    Tang, Wee Kee
      128  109
  • Publication
    Open Access
    On topological Rudin's Lemma, well-filtered spaces and sober spaces
    (2020)
    Xu, Xiaoquan
    ;
    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.
    WOS© Citations 19Scopus© Citations 19  124  68
  • Publication
    Open Access
    A complete Heyting algebra whose Scott space is non-sober
    (2021)
    Xu, Xiaoquan
    ;
    Xi, Xiaoyong
    ;
    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.
    WOS© Citations 15Scopus© Citations 16  95  75
  • Publication
    Open Access
    Lattices of Scott-closed sets
    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.
      347  133
  • Publication
    Restricted
    Gauges of Baire class one functions
    (2007)
    Tang, Wee Kee
    ;
    Zulijanto Atok
    ;
      346  21