Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite 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 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2012
    • (edited Jan 31st 2012)

    just for completeness, I have created an entry almost connected topological group.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2012
    • (edited Feb 2nd 2012)

    It says the “quotient topological space G/G 0G/G_0” and it means the quotient topological group. I know, you wanted to say the underlying space of the quotient topological group, but if one talks about quotient topological space then this means that only G 0G_0 is contracted to a point, while the rest stays the same. In other words, the notation G/G 0G/G_0 in topological category is much bigger than the underlying space of G/G 0G/G_0 in the category of topological groups. The quotient in the category of topological spaces is different than the underlying topological space of a quotient in the category of topological groups. I corrected the statement.

    But a real question is if the entry assumes Hausdorfness. I wonder how one can have compact quotient of topological groups by a connected component if not finite, as if something is an accumulation point, by the homogeneity, everything would be an accumulation point. Example ?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 2nd 2012

    Zoran, as an example, consider the pp-adic integers under addition. This is totally disconnected, so the connected component G 0G_0 is just the identity element. Therefore G/G 0GG/G_0 \cong G, and GG is compact Hausdorff, so this is an example.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2012

    Thanks.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2012

    This example is mentioned already at maximal compact subgroup. But it would be nice if somebody found the time to add it also to almost connected topological group, highlighting the point just discussed.

    (I guess I could do it later. But I won’t protest if somebody does it before me ;-)

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 2nd 2012

    The example if pp-adic integers isn’t in either article. And I can’t decide where to put it in either article.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2012
    • (edited Feb 2nd 2012)

    And I can’t decide where to put it in either article.

    Maybe we should splitt off an entry p-adic integer from p-adic number anyway. The non almost-connectedness would naturally be discussed there and then we could point to that from the other two entries.