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Morgan Rogers (he/him) (Sep 24 2025 at 16:52):I am cross-posting what ended up being a reference request on mathoverflow:
https://mathoverflow.net/questions/500787/when-are-the-points-of-a-topological-space-complemented-as-sublocales
I discovered a (to me) surprising characterization of the separation axiom as saying that singleton subspaces are complemented as sublocales. Has anyone seen this characterization? Have I made an obvious (or subtle) mistake in figuring this out?
Federica Pasqualone (Sep 25 2025 at 06:36):Hi Morgan, if I am not mistaken there is some characterization in Chapter VI of Picado and Pultr (Frames and Locales, Topology without points, Birk.) pages 99-100 and Remark 1.2.1. In this chapter X is assumed to be T_0.
Morgan Rogers (he/him) (Sep 25 2025 at 08:16):@Federica Pasqualone thanks, I've added that to the question, but it goes in the 'easy' direction ' implies such-and-such' whereas I'm looking for a result in the other direction. An equivalent result would be that finite subspaces are complemented as sublocales. In fact, I think I could extend this to "countable subspaces are complemented as sublocales" using the localic Baire category theorem.
Federica Pasqualone (Sep 25 2025 at 09:01):Glad to help! @Morgan Rogers (he/him) I read the book a long time ago, but there is also a last part to the remark 1.2.1 that may be of interest. Cit, "But the converse does not hold, that is, does not make generally a complement of Recall III.8.3: if both A and B are dense they meet at least B in ."
Graham Manuell (Sep 25 2025 at 09:10):@Morgan Rogers (he/him) I am not aware of the characterisation and from a quick look the proof seems fine to me.