Results 1 to 4 of 4

Math Help - Topology Proof

  1. #1
    Newbie
    Joined
    Feb 2008
    Posts
    21

    Topology Proof

    I need some help with a topological proof, hopefully there is someone out there that can help.

    Prove that a bijection f: X→Y is a homeomorphism if and only if f and f^-1 map closed sets to closed sets.


    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2008
    Posts
    148
    The definition of a homeomorphism is a map which is bijective continuous and open (maps open sets to open sets).

    Assume f: X \rightarrow Y is a homeomorphism.

    f is continuous and open so f^{-1} is continuous and open. But then f is closed (maps closed sets to closed sets) and so is f^{-1} since let C be a closed subset of Y. Then C^c is open in Y and so f^{-1}(C^c) is open in X. But f^{-1}(C^c)=(f^{-1}(C))^c and therefore f^{-1}(C) = (f^{-1}(C)^c)^c is closed in X. Same argument shows f is a closed map.

    Now assume that f and f^{-1} are bijective maps which take closed sets to closed sets. We need to show that f is continuous and open. Consider an arbitrary open set U \subset Y. U^c is closed and f^{-1}(U^c) = f^{-1}(U)^c is closed in X. Therefore f^{-1}(U) is open in X. So f is continuous. Similar argument shows f is open.
    Last edited by iknowone; March 25th 2008 at 04:37 PM. Reason: remove a misleading thought
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2008
    Posts
    21
    THanks, but i have one question are C^c and U^c the compliments of C and U? Just want to make sure.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by reagan3nc View Post
    THanks, but i have one question are C^c and U^c the compliments of C and U? Just want to make sure.
    Yes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Understanding Topology Proof
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: July 19th 2011, 08:49 AM
  2. Topology Proof
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 12th 2011, 03:43 PM
  3. Topology proof
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 11th 2009, 07:20 PM
  4. Topology Proof?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 7th 2007, 05:27 PM
  5. Help with Proof intro topology
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 1st 2007, 07:28 PM

Search Tags


/mathhelpforum @mathhelpforum