{"id":2915,"date":"2020-11-21T11:26:33","date_gmt":"2020-11-21T10:26:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2915"},"modified":"2022-11-19T15:00:38","modified_gmt":"2022-11-19T14:00:38","slug":"quasi-uniform-spaces-ii-stably-compact-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2915","title":{"rendered":"Quasi-uniform spaces II: Stably compact spaces"},"content":{"rendered":"\n

Let us continue our exploration of quasi-uniform spaces<\/a>. I was recently reading a paper by Jimmie Lawson [2] about stably compact spaces, and which, I must confess, I did not know about until a few months ago. It contains a wealth of information on stably compact spaces, giving a lucid and comprehensive view of the area. If I could rewrite Chapter 9 of the book<\/a>, I would definitely take example on this paper.<\/p>\n\n\n\n

And, in Section 3.5 of [2], surprise! Jimmie mentions a neat result about the quasi-uniformities that induce the topology of a stably compact space, which he attributes to K\u00fcnzi and Br\u00fcmmer [3]. I immediately decided to talk about it here.<\/p>\n\n\n\n

Let me say what this is about. In the setting of uniform spaces, it has been well-known for a long time that the topology of a compact Hausdorff space is induced by a unique uniformity [1, \u00a74, 1, th\u00e9or\u00e8me 1]. Jimmie says that this result extends to stably compact spaces, and I tried to understand [3] in order to explain why in a hopefully simpler way. Unfortunately, the expected result, which would be that there is a unique quasi-uniformity that induces any given stably compact topology, is wrong, as I will demonstrate below. We will see what goes wrong, and progressively make our way towards a theorem that indeed generalizes the situation on compact Hausdorff spaces.<\/p>\n\n\n\n

Is the topology of stably compact space induced by a unique quasi-uniformity?<\/h2>\n\n\n\n

That is the natural conjecture we might have that would generalize the aforementioned theorem on compact Hausdorff spaces. A quick reading of [2] or [3] might convince you that this what K\u00fcnzi and Br\u00fcmmer somehow proved, but that would be wrong.<\/p>\n\n\n\n

Here is a simple counter-example\u2014in fact almost the same counter-example as last time<\/a>. Consider the interval [0, 1] with the Scott topology of \u2264, whose open intervals are the empty set, the whole interval, and the half-open intervals ]a<\/em>,1], with 0<a<\/em><1. This is a stably compact space. But there are at least two quasi-uniformities that induce the Scott topology:<\/p>\n\n\n\n