Results 1 to 2 of 2

Thread: Homeomorphic finer topology

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    63

    Homeomorphic finer topology

    Find two different topologies on $\displaystyle \mathbb{R}$ where one is strictly finer, yet the two are homeomorphic
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by Andreamet View Post
    Find two different topologies on $\displaystyle \mathbb{R}$ where one is strictly finer, yet the two are homeomorphic
    $\displaystyle (Z, T_1)$ and $\displaystyle (Q, T_2)$ as subspaces of R with a discrete topology.


    Let $\displaystyle B_1$ and $\displaystyle B_2$ be bases for topologies $\displaystyle T_1$ and $\displaystyle T_2$.

    $\displaystyle T_2$ is strictly finer than $\displaystyle T_1$, since for each x in Z and for each basis element $\displaystyle \{x\} \in B_1$ containing x, we have a basis element $\displaystyle \{x\}$ in $\displaystyle B_2$ that contains x. The converse is not necessarily true.

    Since there is a bijection between Z and Q, we have a homeomorphism between $\displaystyle (Z, T_1)$ and $\displaystyle (Q, T_2)$ as subspaces of R with a discrete topology.
    Last edited by aliceinwonderland; Feb 24th 2009 at 10:43 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. R^2 and R^2 - {(0,0)} are not homeomorphic
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Apr 6th 2011, 09:13 AM
  2. product topologies and the concept of finer
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 2nd 2010, 08:22 AM
  3. Homeomorphic spaces
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Feb 21st 2010, 09:18 AM
  4. homeomorphic quotient spaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 21st 2009, 07:51 AM
  5. Help with coarser/finer topologies
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Aug 6th 2008, 06:10 AM

Search Tags


/mathhelpforum @mathhelpforum