Stable Compactification I

This paper represents a continuation of our programme [16, 13] of extending various concepts of general topology from the setting of Hausdorff (or, at most, 7̂ ) spaces, in which they are usually embedded, to the larger classes of spaces we need to consider in the theory of computation. The topic of compactification poses an obvious challenge to this programme, since only a Tychonoff space can have a compactification in the usual sense. This assumes that a compactification has to be a dense embedding into a compact Hausdorff space. By modifying one part or another of this requirement, less restricted constructions can be obtained. Thus, Johnstone's version [11] (following Banaschewski and Mulvey [2]) of the Stone-Cech applies to arbitrary spaces/locales; but this is at the cost of no longer requiring that a compactification be an embedding. In another direction, by admitting non-Hausdorff (quasi-) compact spaces into consideration, Csaszar [4] is able to provide versions of both the Alexandroff and Wallman compactifications for arbitrary spaces. These are remarkable constructions; but their usefulness may be open to question, since compactness on its own is a rather weak property in the case of general spaces. Thus, any space having a least element in its specialization order is trivially compact (and compactification, on Csaszar's account, will, if applicable, leave such a space unchanged). It seems appropriate to look for a stronger notion than that of compactness, yielding a class of spaces that can play a similar role with regard to the general (say, sober) spaces of that of the ordinary compact spaces among the Hausdorff spaces. The most promising candidate for such a strengthened notion is stable compactness (equivalent to stable local compactness of [11]; see comments following Definition 1 below). Accordingly, our proposal is to construct stable compactifications of general spaces. We recall here a few basic definitions and results.