Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 3rd 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs there a place in the general framework of the page [[separation axioms]] for the notion of $T_D$-space (apparently sometimes called $T_{\frac{1}{2}}$?

   Is there a place in the general framework of the page separation axioms for the notion of -space (apparently sometimes called ?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 3rd 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThis looks like a job for Toby! :-)

   This looks like a job for Toby! :-)

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 5th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexNot 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 $T_D$ is on neither list). By the way, I have heard of another separation axiom called ‘$T_{1/2}$’, 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 $T_D$-spaces.

   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.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 5th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI've started reading the new book on locales by Picado and Pultr. They point out that the $T_D$ 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 $F$ such that there exists $a,b$ with $a\in F$, $b\notin F$, and $b\lt a$ with nothing in between. They also give an example of two spaces $X$ and $Y$ with isomorphic frames of opens, such that $X$ is $T_D$ but not sober, while $Y$ is sober but not $T_D$. ($X$ is the cofinite topology on $\mathbb{N}$; $Y$ 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 $T_0$) is less of a "separation" property and more of a "completeness" property.

   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.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 7th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYes, I'm also reading that book. I follow their discussion of $T_D$, 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.

   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.