Results 1 to 3 of 3

Math Help - Bases in Locally Euclidean spaces

  1. #1
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Bases in Locally Euclidean spaces

    Hey everyone, I think I am making the following WAY too difficult. Does anyone have insight into A) whether the following is even correct or B) an easier way to do it (as I am sure there is one).

    Problem: Let \mathfrak{M} be a locally Euclidean space of dimension n. Call an open subset E of \mathfrak{M} a Euclidean ball if E\approx O for some open O\subseteq\mathbb{R}^n. Prove that \mathfrak{M} has an open base of Euclidean balls.

    Proof: Let x\in X be arbitrary and let E_x be the guaranteed Euclidean ball containing x with \varphi_x:E_x\to O_x the necessary homeomorphism. Let \Omega_x=\left\{B_{\varepsilon}(\varphi(x))):B_{\v  arepsilon}\subseteq O_x\right\} (each of these balls are open in \mathbb{R}^n for obvious reasons) and \Lambda_x=\left\{\varphi_x^{-1}(\omega):\omega\in\Omega_x\right\}. Clearly \Lambda_x contains open subset of E_x, but since E_x is in fact open we see that each element of \Lambda_x is open in X. So, set \Lambda=\bigcup_{x\in X}\Lambda_x. By previous comment this is a class of open subsets of \mathfrak{M}. Also, \varphi_x\mid_{\varphi_x^{-1}(\omega)}:\varphi_x^{-1}(\omega)\to\omega is a homeomorphism and so \Lambda is in fact a collection of Euclidean balls. It remains to show that it's a base.

    So, let y\in X be arbitrary and N any neighborhood of y. Thus, keeping the notation of the previous paragraph we have that E_y\cap N is an open subspace of E_y and so \varphi_y(E_y\cap N) is an open subset of O_y. So, there exists some B_{\varepsilon}(\varphi_y(y))\subseteq \varphi_y(E_y\cap N). It follows that \varphi_y^{-1}\left(B_{\varepsilon}(\varphi_y(y))\right)\in\La  mbda and x\in\varphi_y^{-1}\left(B_{\varepsilon}(\varphi_y(y))\right)\subse  teq E_y\cap N\subseteq N. The conclusion follows. \blacksquare.


    Comments?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    Seems right to me, and I don't see anything that's redundant.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Tinyboss View Post
    Seems right to me, and I don't see anything that's redundant.
    Thanks friend!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Separable metric spaces and bases
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 11th 2011, 11:52 PM
  2. [SOLVED] locally compact spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2011, 04:28 AM
  3. [SOLVED] Bases for row and column spaces
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 14th 2011, 04:51 AM
  4. Bases over vector spaces.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 22nd 2010, 06:41 AM
  5. Euclidean Spaces
    Posted in the Calculus Forum
    Replies: 7
    Last Post: October 11th 2008, 08:54 AM

Search Tags


/mathhelpforum @mathhelpforum