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[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
Graham Manuell has marked this topic as unresolved.
I don't know if this has been proved in the literature, but if is compact Hausdorff, isn't the Vietoris locale of the patch topology of ? This would imply that it is compact Hausdorff.
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.
@Graham Manuell - and anyone else who is interested - so I was not aware that the Vietoris on can be viewed as the patch of (for compact Hausdorff ). That's nice and, indeed, in the classical literature as you pointed out separately. I think I have a neat general proof:
Let be a set (=discrete locale). Then (e.g. Vickers). Recalling that embeds (by construction) in , say via then we can construct:
recalling that is discrete for the first bit (so that ). But you can unwind to see (I hope!) that sends any open of to its upper closure (using the order on ). The split must be the opens of and of (the opens of ) , so we have for discrete .
Now, of course, this is an incredibly long winded way of proving the incredibly well know and basic fact: the power set of , where is the set of finite subsets of . But the point of knowing a purely localic proof of for discrete is the we can then appeal to compact Hausdorff/discrete duality: for compact Hausdorff . And since we know we get that as patch reverses ideal completion!
@Christopher Townsend Nice. How do you get the first isomorphism in the definition of formally?
Graham Manuell said:
Christopher Townsend Nice. How do you get the first isomorphism in the definition of formally?
is discrete - so apply for any discrete *. That is discrete for discrete follows by applying (formally) the standard join-coverage result using that the finite diagonals on are open. (This was really the original question: is this step written down for the nullary diagonal ? I now think it sort of is in Vickers' paper on the connected Vietoris, though not 'formal' there, easily adapted to become formal. ).
(Note: I probably should not have used the notation when talking about the compact Hausdorff dual to discrete ...)
I meant how do you get is discrete in a way that dualises to the compact Hausdorff case, but yes, as you say, I guess that was the original question.