Completeness of Quasi‐Uniform and Syntopological Spaces

In this paper we begin to develop the filter approach to (completeness of) quasiuniform spaces, proposed in [8, Section V]. It will be seen that this permits a more powerful and elegant account of completion to be given than was feasible using sequences or nets [8]. Just as in the previous version [8], we find that received notions concerning convergence need to be revised and reformulated to deal adequately with the nonsymmetric (non-Hausdorff) situation. The detailed motivation for the revisions concerning filter convergence (Sections 1, 4, 5 below) is independent of that given previously [8] for revising the notions of convergence of sequences and nets; the fact that the two sets of revisions lead to results that are in good agreement with each other tends to confirm the soundness of the revisions. The filter approach is more general in scope than the sequence/net approach. In particular, we shall see below that sobrification is a special case of the filter completion construction. The connection with sober spaces and locales is certainly one that can be pursued further, and indeed we intend the work reported here as at least a step in the direction of a localic, point-free, or even information system (in the sense of Scott) view of quasi-uniformities. Although all the examples and applications we have in mind are quasi-uniform spaces, we have found it helpful to formulate much of the material in terms of a still more general concept, namely the syntopological spaces of Csaszar [2]. We have in fact used syntopological formulations in a piecemeal fashion in previous versions [8], but we shall be doing this more systematically here.