The inconvenient category of topological spaces

Lewis, Ronald S. (2003) The inconvenient category of topological spaces. Masters thesis, Memorial University of Newfoundland.

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Abstract

An extensive classification of initial and final topologies characterizes chapters one through three. The one exception is a generalization of the locally compact and Hausdorff concept which appears at the beginning of chapter three and plays a role of significance later in the thesis. Most of work in the first three chapters is standard material, the general theory is laid out and followed by specific constructions. Features of this treatment include an initial topology in the function space setting, several final topology constructions that satisfy convenient category criteria, and some basic properties of a specific product topology are explored in detail. Modification of the exponential law is the thrust of chapter four. There is a desire for a law which utilizes no assumptions on the spaces involved. A compact Hausdorff image-open topology is defined to replace the standard compact-open topology on a function space. The χ-open topology coupled with a χ-product topology give life to a χ-exponential law that has the usual exponential law as a consequence. A fifth chapter examines initial and final topologies in regards to their commutativity. Improvements to the current body of knowledge are made in the area of product and identification commutative. In particular, an interesting case of initial and final commutation using fibred mapping spaces is explored.

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