11 research outputs found
Bi-Alexandroff spaces
No full textBi-Alexandroff spaces are defined as extensions of Alexandroff spaces [1]. Urysohn’s lemma for bi-Alexandroff spaces is used to show that upper and lower cozero sets of bitopological spaces are bi-Alexandroff spaces. An adjunction between bi-Alexandroff spaces and pairwise completely regular bitopological spaces is established
Stably Continuous σ- Frames
No full textIt is shown that the categories of stably continuous σ-frames
and compact regular σ-biframes are equivalent. This is the analogue
of Banaschewski and Brümmer [1], linking the stably continuous frames and compact
regular biframes. Mathematics Subject Classification (1991): 54D30, 54D20, 06B35, 06D05 Keywords: stably continuous s-frames, compact regular s-biframes, compactness,
noncompact covering properties, continuous lattices,generalizations,applications,
structure and representation theory, categories, category, stably continuous
frame, frames, frame Quaestiones Mathematicae 24(2) 2001, 201-21
Pointfree pseudocompactness revisited
No full textAbstractWe give several internal and external characterizations of pseudocompactness in frames which extend (and transcend) analogous characterizations in topological spaces. In the case of internal characterizations we do not make reference (explicitly or implicitly) to the reals
(L, M)-fuzzy topological spaces
No full textThe objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices
Implementing innovative assessment methods in undergraduate Mathematics
No full textThe following challenges associated with teaching undergraduate Mathematics will be discussed: Negative attitudes of students to Mathematics, Student’s reluctance to practise Mathematics, and surface learning. Some (or all) of the ways in which assessment can be used to address these challenges will be discussed. If used strategically assessment methods/tasks can enhance the teaching and learning of mathematics. Some of the unique challenges that we as lecturers face in teaching mathematics can be remedied by selecting appropriate assessment techniques/tasks. Using the tutorial time fruitfully is one of the challenging aspects in teaching mathematics. Not taking tutorials and other formative assessments seriously is not an uncommon student attitude in higher education contexts. Students who are pressured for time often do not see the immediate value of formative assessment or of discussion as a useful learning activity. A collection of case studies which clearly document what has been tried in different contexts is very useful in mathematics as this information is limited in the South African higher education sector. An innovative assessment method (peer-assessment) which was introduced for a Linear Algebra second year course at Rhodes University (South Africa) will be presented: The implementation method, purpose of introducing the assessment method, its advantages, and disadvantages will be examined. A reflection on the assessment method and concluding remarks will be provided
A FEW POINTS ON POINTFREE PSEUDOCOMPACTNESS
No full textAbstract. We present several characterizations of completely regular pseudocompact frames. The first is an extension to frames of characterizations of completely regular pseudocompact spaces given by Väänänen [19]. We follow with an embedding-type characterization stating that a completely regular frame is pseudocompact if and only if it is a P-quotient of its Stone-Čech compactification. We then give a characterization in terms of ideals in the cozero parts of the frames concerned. This characterization seems to be new and its spatial counterpart does not seem to have been observed before. We also define relatively pseudocompact quo-tients, and show that a necessary and sufficient condition for a completely regular frame to be pseudocompact is that it be relatively pseudocompact in its Hewitt realcompactification. Consequently a proof of a result of Banaschewski and Gilmour [6] that a completely regular frame is pseudocompact if and only if its Hewitt realcompactification is compact, is presented without the invocation of the Boolean Ultrafilter Theorem. This paper is, in a way, a sequel to our paper [11] in which we gave several characterizations of pseudocompact frames with no separation axiom imposed. In this paper we restrict to completely regular frames. We start by extending to frames Väänänen’s [19] characterization
Bi-Alexandroff spaces
No full textBi-Alexandroff spaces are defined as extensions of Alexandroff spaces [1]. Urysohn’s lemma for bi-Alexandroff spaces is used to show that upper and lower cozero sets of bitopological spaces are bi-Alexandroff spaces. An adjunction between bi-Alexandroff spaces and pairwise completely regular bitopological spaces is established
A few points on Pointfree pseudocompactness
No full textWe present several characterizations of completely regular pseudocompact frames. The first is an extension to frames of characterizations of completely regular pseudocompact spaces given by Väänänen. We follow with an embedding-type characterization stating that a completely regular frame is pseudocompact if and only if it is a P-quotient of its Stone-Čech compactification. We then give a characterization in terms of ideals in the cozero parts of the frames concerned. This characterization seems to be new and its spatial counterpart does not seem to have been observed before. We also define relatively pseudocompact quotients, and show that a necessary and sufficient condition for a completely regular frame to be pseudocompact is that it be relatively pseudocompact in its Hewitt realcompactification. Consequently a proof of a result of Banaschewski and Gilmour that a completely regular frame is pseudocompact if and only if its Hewitt realcompactification is compact, is presented without the invocation of the Boolean Ultrafilter Theorem
