Thesis (M.A.)--University of Hong Kong, 1967. / Also available in print.

The formulation of the Baire category theorem found in most elementary topology texts deals with two distinct classes of spaces: locally compact spaces, and complete metric spaces. This "dual theorem" status of Baire's theorem suggests the problem of finding one class of topological spaces for which the Baire category theorem can be proved and which includes both the locally compact spaces and the complete metric spaces. This thesis surveys and compares the three approaches to this problem taken by three methamticians. The classical results of E. Čech achieve a unified Baire theorem by a Aefinl.ti.on of completeness different from that in current common usage. Johannes de Groot introduced a notion of subcompactness, generalizing compactness. K. Kunugi worked in the setting of complete ranked spaces which generalize uniform spaces and eliminate the need to assume regular separation in the space. This last point is the basis for the construction of a complete ranked space which is neither subcompact nor complete in the sense of Čech. It is also shown in the paper that there exist spaces subcompact but not complete in the sense of Čech, and that in certain special cases completeness in the sense of Čech implies subcompactness. / Science, Faculty of / Mathematics, Department of / Graduate

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Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: While the notions of open and closed subsets in a topological space are dual to each other, they take on another meaning when points and complements are no longer available. Closure operators have been extensively used to study topological notions on categories. Though this has recovered a fair amount of topological results and has brought an economy of e ort and insight into Topology, it is thought that certain properties, such as convergence, are naturally associated with neighbourhoods. On the other hand, it is interesting enough to investigate certain notions, such as that of closed maps, which in turn are naturally associated with closure by means of neighbourhoods. We propose in this thesis a set of axioms for neighbourhoods and test them with the properties of connectedness and compactness. / AFRIKAANSE OPSOMMING: Al is die twee konsepte van oop en geslote subversamelings in 'n topologiese ruimte teenoorgesteldes van mekaar, verander hul betekenis wanneer punte en komplemente nie meer ter sprake is nie. Die gebruik van afsluitingsoperatore is alreeds omvattend in die studie van topologiese konsepte in kategorieë, toegepas. Alhoewel 'n redelike aantal topologiese resultate, groeiende belangstelling en groter insig tot Topologie die gevolg was, word daar geglo dat seker eienskappe, soos konvergensie, op 'n natuurlike wyse aan omgewings verwant is. Nietemin is dit van belang om sekere eienskappe, soos geslote afbeeldings, wat natuurlik verwant is aan afsluiting, te bestudeer. In hierdie proefskrif stel ons 'n aantal aksiomas oor omgewings voor en toets dit gevolglik met die eienskappe van samehangendheid en kompaktheid.

Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaf 49). Also available on the World Wide Web.