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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2012

    Is there a place in the general framework of the page separation axioms for the notion of T DT_D-space (apparently sometimes called T 12T_{\frac{1}{2}}?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 3rd 2012

    This looks like a job for Toby! :-)

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 5th 2012

    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 T DT_D is on neither list). By the way, I have heard of another separation axiom called ‘T 1/2T_{1/2}’, 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 T DT_D-spaces.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 5th 2012

    I’ve started reading the new book on locales by Picado and Pultr. They point out that the T DT_D 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 FF such that there exists a,ba,b with aFa\in F, bFb\notin F, and b<ab\lt a with nothing in between. They also give an example of two spaces XX and YY with isomorphic frames of opens, such that XX is T DT_D but not sober, while YY is sober but not T DT_D. (XX is the cofinite topology on \mathbb{N}; YY 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 T 0T_0) is less of a “separation” property and more of a “completeness” property.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 7th 2012

    Yes, I’m also reading that book. I follow their discussion of T DT_D, 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.