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Unifying the Baire category theorem Huber, George Michael

Abstract

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.

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