Results 1 to 3 of 3

Math Help - property of a topological group??

  1. #1
    Junior Member
    Joined
    Feb 2011
    Posts
    72

    property of a topological group??

    Hello;

    Show that if G is a first countable topological group, then there is a sequence of symmetric neighborhoods (U_{n})_{n} of the neutral element e of G such that

    (1){ {U_{n}:n\in Z }} is a local base at e in G


    (2) U_{n+1}^{3}\subset U_{n}, for every n

    I understand number 1 is due to first countability of G, but how can we get (2). Please guide me. Every comment or guidance is highly appreciated

    Recall that a neighborhood of the identity element is said to be symmetric if it equal to its inverse.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: property of a topological group??

    Did you try to use the continuity of the map f\colon G^3\to G, f(x,y,z)=xyz?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2011
    Posts
    72

    Re: property of a topological group??

    Ok, thanks a lot my instructor. Now, I understand the concept.

    Let U be any neighborhood of the identical element e in G. By continuity of (x,y,z)\Rightarrow xyz there is a neighborhood W_{1} of e such that W_{1}^{3}\subset U. By continuity of (x,y) \Rightarrow xy^{-1} there is a neighborhood W_{2} of e such that W_{2}W_{2}^{-1} \subset W_{1}. That's mean any neighborhood of e contains a symmetric neighborhood. Now V=W_{2}W{2}^{-1} is a symmetric neighborhood of e and V^{3} \subset W_{1}^{3} \subset U. Because of that, we can construct such a sequence of symmetric neighborhood of e.

    Thank you very much for the very helpful guidance.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Topological group/locally compact subgroup
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: November 6th 2010, 06:59 PM
  2. [SOLVED] A question in topological group
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: September 17th 2010, 12:55 AM
  3. Proof of Topological Property
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 3rd 2010, 03:18 PM
  4. Abelian Group Property
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 8th 2009, 06:53 PM
  5. Topological property
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 30th 2007, 02:58 PM

Search Tags


/mathhelpforum @mathhelpforum