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UBC Theses and Dissertations

Diagonal spaces Beckmann, Philip Valentine

Abstract

In their monograph 'Quasi-uniform Topological Spaces’, M.G-. Murdeshwar and S.A. Naimpally documented those "uniformity" results which carry over to quasi-uniformities, a quasi-uniformity being a uniformity that lacks the symmetry property. It seemed natural to ask what results might remain true if the "triangle" property of a uniformity were also removed. The investigation of this idea here has resulted in a rather primitive (from a topological point of view) structure called a "diagonal space". Unfortunately, since a topology can not be obtained in the usual way from such a diagonal structure, most of the desirable standard results do not carry over. In Chapter 0 the basic notions that are needed are defined, the most important being that of a filter. Chapter 1 deals with diagonal spaces and the pseudo-topologies that they generate. The latter part of Chapter 1 outlines techniques whereby a pseudo-topology can be "reduced" to a topology. The relationship between diagonal filters and the "pretopologies" of D.C. Kent is discussed in Chapter 2 along with the various relationships between the topologies and generalizations of topolgies that can be defined in a natural way from diagonal spaces and pretopologies. Finally, in Chapter 3, there is a very brief discussion on the analogues, in terms of diagonal spaces and pretopologies, of a few standard concepts of topology.

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