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Dec 21, 2019 at 19:46 comment added Ivan Di Liberti Is there any connection between these guys you are studying and the bitopological description of Priestley spaces?
Dec 18, 2019 at 18:07 comment added Tim Campion @LSpice Eh, you can view it as just being like a prime. It started because I was thinking of a $\beta$ structure in terms of the structure map $\xi: \beta X \to X$, but then the formulation I arrived at didn't mention this map at all. So it's kind of a relic.
Dec 18, 2019 at 18:06 comment added LSpice Is the $\xi$ in $\tau^\xi$ just punctuation giving a new name, like a prime $\tau'$, or does it have some extra meaning?
Dec 18, 2019 at 18:05 history edited LSpice CC BY-SA 4.0
Name of [here], and \operatorname's
Dec 18, 2019 at 16:24 answer added Andrea Marino timeline score: 2
Dec 18, 2019 at 14:40 comment added Tim Campion @AndrejBauer It certainly looks like that, which is partly why I'm hopeful that it's a condition which may have been studied before...
Dec 18, 2019 at 14:39 comment added Tim Campion @AndreaMarino I'm confused -- surely you're not saying that for any topology $\tau$ there is at most one $\tau^\xi$ which defines a $\beta$-space (since any infinite indiscrete space is a counterexample)? I'd be very interested to hear more about this!
Dec 18, 2019 at 12:23 comment added Andrea Marino Well I found a pretty indirect characterization. There exist an explicitly constructible refinement $\tau'$, and $\tau$ defines a $\beta$ space iff $\tau'$ is compact and locally compact. If you want I can post it, but I think we can do better. A necessary condition though is that $X$ must be also locally compact .
Dec 18, 2019 at 8:19 comment added Andrej Bauer Condition (3) looks like some sort of regularity of $\tau$ relative to $\tau^\xi$, for if $\tau = \tau^\xi$ it's just ordinary regularity, isn't it?
Dec 18, 2019 at 1:42 comment added Tim Campion @AndreaMarino I agree! I think it's worthwhile to revisit some of these classic things from time to time. Manes' theorem, in particular, is a gem which deserves to be more widely known. The proof -- once you know what the Stone-Cech compactification of a discrete space is -- is an easy, fun application of the Beck Monadicity Theorem. And there's a whole cottage industry of extensions of these ideas, starting with a description of an arbitrary topological space as a kind of "lax algebra" for the ultrafilter monad.
Dec 18, 2019 at 1:12 comment added Andrea Marino This is so cool. I missed good point topology problems :)
Dec 18, 2019 at 0:06 history edited Martin Sleziak
added the (monads) tag - feel free to rollback my edit if I missed something and the tag is not suitable here
Dec 17, 2019 at 23:43 history edited Tim Campion CC BY-SA 4.0
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Dec 17, 2019 at 18:43 history edited Tim Campion CC BY-SA 4.0
deleted 59 characters in body
Dec 17, 2019 at 18:35 history asked Tim Campion CC BY-SA 4.0