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\(p\)-regular Cauchy completions. (English) Zbl 0996.54034

Introduction: Since a paper of \(p\)-regular completions [D. C. Kent and G. D. Richardson, Math. Nachr. 151, 263-271 (1991; Zbl 0762.54021)], some recent and relevant developments have occurred which provide the motivation for this paper. The discovery of a dual relationship between the properties “\(p\)-regular” and “\(p\)-topological” in convergence space theory led to an investigation of \(p\)-topological convergence spaces [S. A. Wilde and D. C. Kent, Int. J. Math. Math. Sci. 22, No. 1, 1-12 (1999; Zbl 0936.54003)] and \(p\)-topological Cauchy completions [J. Wig and D. C. Kent, ibid., No. 3, 497-509 (1999; Zbl 0979.54030)]. These, in turn, raised some questions about \(p\)-regular completion theory that had not been previously considered, such as the possible duality between \(p\)-regular and \(p\)-topological spaces, the role of diagonal axioms in the study of \(p\)-regular completions, the existence of \(p\)-topological Reed completions, and the relevance of \(p\)-regular completions to the study of regular completions. These topics, along with a comparison of the behaviour of \(p\)-regular and \(p\)-topological completions, form the subject matter of this paper.

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54E15 Uniform structures and generalizations
54A05 Topological spaces and generalizations (closure spaces, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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