##
**\(p\)-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.) |