On Smallest Compactification for Convergence Spaces

In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification. Introduction. A Hausdorff convergence space as defined in [1] always has a Stone-Cech compactification which can be obtained by a slight modification of the result in [3]. But in general this need not be the largest Hausdorff compactification of the space, and in fact it has been pointed out in [4] that the number of distinct maximal Hausdorff compactifications can be quite large. In this note we define the notion of local compactness for a Hausdorff convergence space and show that a Hausdorff noncompact convergence space has a smallest Hausdorff compactification iff the space is locally compact. With a view to obtain a more satisfactory compactification theory for convergence spaces, Richardson and Kent have considered regular compactifications in [4] and have obtained a characterization of the class of convergence spaces for which regular compactifications exist and have shown that each such convergence space has a largest regular compactification. In this note such a convergence space is called an ^-convergence space, and it has been shown that an R-convergence space has a smallest regular compactification iff its pre-topological modification is a locally compact topological space. 1. For terms and results about convergence spaces used in this paper, we refer to [1] and [4]. A convergence space (5, q), where q is the convergence structure will be denoted simply by S, and //-convergence and //-adherence points will be referred to as .^-convergence and 5-adherence points respectively, x will denote the principal ultrafilter generated by {x}. For a filter J^ on F, if its trace on a subset S of Fexists, will be denoted by J^nS, and the filter generated by ̂ nS on F will be denoted by [J^n S]. S will be called F-open if S belongs to every filter on F that F-converges Received by the editors May 10, 1973 and, in revised form, July 6, 1973. AMS (MOS) subject classifications (1970). Primary 54A05, 54A20; Secondary 54D30, 54D35, 54D45.