Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Is there a place in the general framework of the page separation axioms for the notion of -space (apparently sometimes called ?
This looks like a job for Toby! :-)
Not as far as I know.
Not every separation axiom fits into that scheme; see http://ncatlab.org/nlab/show/separation%20axioms#other_axioms_7 and https://en.wikipedia.org/wiki/Separation_axiom#Other_separation_axioms for places to list exceptions (although is on neither list). By the way, I have heard of another separation axiom called ‘’, but I never tracked down the reference and nothing came of the discussion: https://en.wikipedia.org/wiki/Talk:Separation_axiom#T1.2F2_spaces.3F.
I don’t really have a good understanding of -spaces.
I’ve started reading the new book on locales by Picado and Pultr. They point out that the axiom also allows one to recover a space up to homeomorphism from its frame of open sets, though in a different way from the sober axiom. Rather than recovering points as completely prime filters, as we do for a sober space, we find them as prime filters such that there exists with , , and with nothing in between. They also give an example of two spaces and with isomorphic frames of opens, such that is but not sober, while is sober but not . ( is the cofinite topology on ; is its one-point compactification.)
They also make the point, which seems reasonable to me, that sobriety (or, at least, the half of sobriety that isn’t ) is less of a “separation” property and more of a “completeness” property.
Yes, I’m also reading that book. I follow their discussion of , but I still don’t feel that I really understand the concept.
I also agree that sobriety isn’t really about separation, but then neither is perfect normality. Both should still be on a list of ‘other’ separation axioms, since other people list them sometimes.
1 to 5 of 5