Abstract. We are primarily concerned with minimal P convergence spaces, where P is one of the following convergence space properties: HausdorfT, T2, A-regular, A-Urysohn, and first countable, A an infinite cardinal number. Our conclusions usually resemble the corresponding topological results, but with some deviations ; for instance, a minimal HausdorfT convergence space is always compact, whereas a countable minimal regular convergence space need not be compact.
Abstract.This note gives a simple characterization for the class of convergence spaces for which regular compactifications exist and shows that each such convergence space has a largest regular compactification.
Introduction.It has been shown by Wyler [5] that for every Hausdorff convergence space S there is a regular (including Hausdorff) compact convergence space S* and a continuous map j:S-+S* with the following property: for every continuous map f-.S-^-T, where Pis regular and compact, there is a unique continuous map g:S*^>-T such that f=g°j-Richardson [4] obtained a similar result, but with the following important distinctions: (1) the compactification space S* is Hausdorff but not necessarily regular (for convergence spaces, Hausdorff plus compact does not imply regular); (2) the map j is a dense embedding. But there is in general no largest Hausdorff compactification, and indeed the number of distinct maximal Hausdorff compactifications can be quite large.The conclusions of both [4] and [5] suggest that a more satisfactory compactification theory for convergence spaces might result from an investigation of regular compactifications, although it is known (see [2]) that there are regular convergence spaces which cannot be embedded in any compact regular space. What we obtain in this note is a characterization of the class of convergence spaces for which regular compactifications exist, and we show that each such convergence space has a largest regular compactification.
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