A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional. Subcategories of compact and of complete spaces are investigated. It is shown that each T-convergence space has a compactification with the extension property provided the frame is a Boolean algebra. T-Cauchy spaces are defined and sufficient conditions for the existence of a completion are given. T-uniform limit spaces are also defined and their completions are given in terms of the T-Cauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed T-convergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded T-Cauchy spaces which induce the fixed space.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Reid, Frederick, "Lattice-Valued T-Filters and Induced Structures" (2019). Electronic Theses and Dissertations, 2004-2019. 6334.