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Stream: theory: category theory

Topic: Vietoris Locale Research Question

Christopher Townsend (Aug 05 2025 at 06:24):

[Cross post from Math overflow & categories mailing list.]

Let X be a locale. Then there is the Vietoris locale construction, V, introduced by Johnstone (it's in Stone Spaces, but see also the 1985 paper, 'Vietoris Locales and Localic Semilattices'). The functor V:Loc→Loc is something like taking the locale of all 'finite' sublocales; e.g., if X is discrete then the points of V(X) are the finite subsets of X.  In his 1985 paper Johnstone alludes to the fact that if X is compact then so is V(X). But the proof, which is not explicit in the paper, uses a transfinite induction. I believe there is a much simpler proof, but have not found any references to this in the literature. Does anyone know about this specific result being published in the last 40[!] years?

Thanks,
Christopher

Notification Bot (Aug 08 2025 at 10:25):

Graham Manuell has marked this topic as unresolved.

Graham Manuell (Aug 08 2025 at 10:27):

I don't know if this has been proved in the literature, but if $X$ is compact Hausdorff, isn't the Vietoris locale of $X$ the patch topology of $\mathbb{S}^X$? This would imply that it is compact Hausdorff.

Christopher Townsend (Aug 11 2025 at 18:11):

Thanks Graham. I'd have to think about that specific point. It rings true as some of my thinking has been towards seeing the opens of V(X) as a splitting in the same way that the patch can be constructed as a splitting (interestingly, using a dcpo idempotent for general locales X, for which I think I was going to look at your paper that expresses the localic surjection of the splitting in such a circumstances, and moreover describes the splitting locale's presentation, I think?).

Of course, we know preservation by V:Loc->Loc of compact Hausdorffness from Stone Spaces - it was the preservation of compactness absent anything else that I was after. It's not really a 'big' result - it's a bit like how binary Tychonoff becomes straight forward once you use the preframe presentation of the (opens of) locale product. There's a similar result in Vicker's paper on connected Vietoris about the preservation of openness. I just wanted to see if anyone else had actually published anything on this before I wrote anything up.