Completeness of Quasi‐Uniform and Syntopological Spaces
Completeness of Quasi‐Uniform and Syntopological Spaces
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DOI：
10.1112/jlms/49.2.385
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发表时间：
1994-04
影响因子：
1.2
通讯作者：
M. Smyth
中科院分区：
文献类型：
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作者：
M. Smyth
在本文中，我们开始开发准均匀空间（完整性）的过滤方法，在[8，第五节]中提出。可以看出，与使用序列或网络[8]相比，这允许给出更强大和更优雅的完成说明。正如在之前的版本 [8] 中一样，我们发现需要修改和重新表述有关收敛的公认概念，以充分处理非对称（非豪斯多夫）情况。关于滤波器收敛（下面第 1、4、5 节）的修订的详细动机独立于之前给出的 [8] 修订序列和网络收敛概念的动机；两组修订产生的结果彼此非常一致，这一事实往往证实了修订的合理性。过滤器方法在范围上比序列/网络方法更通用。特别是，我们将在下面看到，清醒是过滤器完成结构的一个特例。与清醒空间和场所的联系当然是可以进一步追求的，事实上，我们希望这里报告的工作至少是朝着局部的、无点的、甚至信息系统（在斯科特的意义上）准均匀性观点的方向迈出的一步。尽管我们想到的所有示例和应用都是准均匀空间，但我们发现根据更一般的概念（即 Csaszar 的合成空间）来表述大部分材料是有帮助的[2]。事实上，我们在以前的版本中已经以零碎的方式使用了句法表述[8]，但我们将在这里更系统地这样做。
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.