Results 1 to 5 of 5

Math Help - Example of a non-Hausdorff surjective function

  1. #1
    Newbie Throughpoint's Avatar
    Joined
    Nov 2010
    Posts
    11

    Example of a non-Hausdorff surjective function

    Let X = [-1,1] and give X the usual topology.

    Give an exmple of a surjective function g from X onto [0,1] such that the quotient topology on [0,1] induced by g is not Hausdorff.

    Thanks for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Throughpoint View Post
    Let X = [-1,1] and give X the usual topology.

    Give an exmple of a surjective function g from X onto [0,1] such that the quotient topology on [0,1] induced by g is not Hausdorff.

    Thanks for any help.
    What have you tried? Note that since X is compact Hausdorff that saying the the topology induced on [0,1] by some quotient map q is equivalent to stating that q is a closed map. So, find a quotient map q:[-1,1]\to[0,1] which isn't closed and the resulting coinduced topology on Y will be non-Hausdorff.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie Throughpoint's Avatar
    Joined
    Nov 2010
    Posts
    11
    So how about..

    g(x) = x^2? Lol. Every element in [0,1] has a pre-image in [-1,-1].

    I might be way off.

    Thanks so much for the help so far btw. ^^

    No wait, how about g(x) = 0 if x is a rational number, or 1 if x is not a rational number.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Throughpoint View Post
    So how about..

    g(x) = x^2? Lol. Every element in [0,1] has a pre-image in [-1,-1].

    I might be way off.

    Thanks so much for the help so far btw. ^^

    No wait, how about g(x) = 0 if x is a rational number, or 1 if x is not a rational number.
    Ok, in your examples what closed set's image isn't closed?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie Throughpoint's Avatar
    Joined
    Nov 2010
    Posts
    11
    Erm.. I guess g = x^2 does have a closed image.. as it maps onto [0,1]. That doesn't mean it's not closed IN [-1,1] does it?

    My other example isn't closed in [-1,1]!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] A subspace of a Hausdorff space is Hausdorff
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: September 24th 2011, 01:40 PM
  2. surjective function
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: March 17th 2011, 06:45 AM
  3. Surjective Function
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 13th 2010, 09:27 PM
  4. Quotient map from Hausdorff to non-Hausdorff space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 28th 2009, 05:51 PM
  5. surjective function
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: August 18th 2008, 06:26 AM

Search Tags


/mathhelpforum @mathhelpforum