Browsing by Author "Gilmour, Christopher Robert Anderson"
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- ItemOpen AccessBisigma frames(1999) Matutu, P; Gilmour, Christopher Robert Anderson; Brümmer, G C LWe introduce and investigate the concept of a bi σ-frame. The cozero part of a biframe, itself a bi σ-frame, is defined and used to construct the compact regular, and regular Lindelöf coreflections for biframes. Pseudocompactness for biframes is defined in a natural way and characterised in terms of the cozero part. Finally we obtain the σ-frame analogue of the result that characterises the stably continuous frames in terms of the compact regular biframes.
- ItemOpen AccessCongruence frames of frames and k-frames(2015) Manuell, Graham Richard; Frith, John; Gilmour, Christopher Robert AndersonWe describe the congruence lattices of frames and k-frames. We look at the role that congruence biframes play in the category of strictly zero-dimensional biframes and discuss some reflections and coreflections of congruence frames.
- ItemOpen AccessE-compactness in pointfree topology(1998) Marcus, Nizar; Gilmour, Christopher Robert AndersonThe main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames.
- ItemOpen AccessFactorization properties of universal algebras(2010) Wiggins, Harry; Ouwehand, Peter; Künzi, Hans-Peter A; Gilmour, Christopher Robert AndersonThis dissertation deals with algebraic structures that can be written as the product of directly indecomposable algebras in a unique way up to isomorphism, known as the Unique Factorization Property. Here we undertake the task of collecting all the major results discovered by a few mathematicians (A. Tarski, B. J´onsson, R. Mckenzie, C. Chang, G. Birkhoff, L. Lovasz, etc.) over the past century. Another goal of this thesis was to highlight important and to introduce fresh techniques. The scope of most of them is still unknown and hopefully they can be utilised further to yield new results to revive this beautiful branch of mathematics.
- ItemOpen AccessNearness and convergence in pointfree topology(2004) Naidoo, Inderasan; Gilmour, Christopher Robert AndersonWe introduce and investigate the concept of a nearness structure on a σ-frame. Analogues of the Samuel Compactification, Uniform Coreflection and Completion in the nearness σ-frame setting are obtained. Convergence in uniform frames is also a subject of this thesis integrating compactness, precompactness and paracompactness. Finally, the notion of uniform paracompactness is introduced and its relation with convergence is investigated.
- ItemOpen AccessNormal bases and compactifications of frames(1995) Robertson, Angela May; Gilmour, Christopher Robert AndersonThe initial aim of this dissertation was to provide a frame-theoretic analogue of Banaschewski's normal systems of sets in [5] as well as a frame counterpart to their associated compactifications. Having completed this part of the task, it seemed natural to investigate the relationship between this compactification and those mentioned above. Hence the first five chapters of the dissertation are devoted to the study of the frame counterparts to six well-known compactifications in the category of topological spaces. For each compactification studied, we give some motivation as to why it should be regarded as a frame-theoretic analogue of its classical counterpart. The sixth chapter is concerned with the relationships between the compactifications: in particular we are interested in conditions under which the different constructions give rise to the same compactification.
- ItemOpen AccessRealcompact Alexandroff spaces and regular σ-frames(1981) Gilmour, Christopher Robert Anderson; Brümmer, Guillaume C LIn the early 1940's, A.D. Alexandroff [1940), [1941) and [1943] introduced a concept of space, more general than topological space, in order to obtain a simple connection between a space and the system of real-valued functions defined on it. Such a connection aided the investigation of the relationships between the linear functionals on these systems of functions and the additive set functions defined on the space. The Alexandroff spaces of this thesis are what Alexandroff himself called the completely normal spaces and what H. Gordon [1971) called the zero-set spaces.
- ItemOpen AccessRealcompactifications of frames(1993) Marcus, Nizar; Gilmour, Christopher Robert AndersonThe first notion of realcompactness in frames was introduced by Reynolds [1979], and it was shown by Madden and Vermeer [1986] that this coincides with the Lindelof property. My thesis advisor suggested that more general realcompactifications of a frame L could be constructed by considering regular sub σ-frames which join generate L. This was motivated by the fact that the Alexandroff bases, which are used to construct the Wallman realcompactifications of a space X, are, as shown by Gilmour, simply the regular sub σ-frames of the frame of open sets of X. The key definition of realcompactness needed here is due to Schlitt [1990] and it is his construction of the universal realcompactification that we modify in order to obtain the Wallman realcompactifications.
- ItemOpen AccessUniform sigma frames and the cozero part of uniform frames(1989) Walters, J L; Gilmour, Christopher Robert AndersonIn this thesis some general results on uniform frames are established and then, after defining a 'uniform sigma frame', the correspondence between the two is explored via the 'uniform cozero part' of a uniform frame. It is shown that the Lindelof uniform frames and the uniform sigma frames are in fact equivalent as categories, and that properties of, and constructions using separable uniform frames can be obtained by considering the uniform cozero part. For example, the Samuel compactification of a separable uniform frame can be obtained via the Samuel compactification (in the sigma frame sense) of the underlying cozero part of the uniform frame. Throughout the thesis, choice principles such as the axioms of choice and countably dependent choice, are used, and generally without mention.
- ItemOpen Accessσ-Spaces and σ-frames(2006) Clarke, Jumani; Gilmour, Christopher Robert AndersonIncludes bibliographical references.