Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1978. / Also available via the Internet.

Thesis (M.S.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographical references (ℓ.54).

Thesis (M.S.)--Tennessee Technological University, 2008. / Title from title page screen (viewed on Sept. 29, 2009). Bibliography: leaves 32-33.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1976. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

A topological space may be viewed as an algebraic structure. For example, it may be viewed as a (complete atomic) Boolean algebra equipped with a closure operator. The lattice of closed subsets is another algebraic structure which may be associated with a topological space. Tne purpose of this thesis is primarily to investigate the metamathematical properties of algebraic structures associated with topological spaces. More specifically, we will first consider questions of decidability of the theories of these algebraic structures. It turns out that these theories are undecidable. We will also examine certain equivalence relations on the class of topological spaces that arise naturally from viewing them as first-order structures. Finally we will show that certain classical theorems of model theory do not hold for topological spaces. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this paper is to further investigate R0 spaces, R1 spaces, and hyperspaces. The R0 axiom was introduced by N. A. Shanin in 1943. Later, in 1961, A. S. Davis investigated R0 spaces and introduced R1 spaces. Then, in 1975, William Dunham further investigated R1 spaces and proved that several well-known theorems can be generalized from a T2 setting to an R1 setting. In Chapter II R0 and R1 spaces are investigated and additional theorems that can be generalized from a T2 setting to an R1 setting are obtained.

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.

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In the study of groups and topological spaces, the properties of both are often encountered in one system. The following are common examples: groups with discrete topologies, the complex numbers with the usual topology, and matrix groups with metric topologies. The need for a study of how algebraic properties and topological properties affect one another when united and interrelated in one system soon becomes evident. Thus the purpose of this thesis is to study the interrelated group and topological space, the topological group.