Results 1 to 2 of 2

Math Help - retract in a topological space

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    98

    retract in a topological space

    If X is a topological space, what would be an example of a retract of X which is not closed?

    I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

    [A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by math8 View Post
    If X is a topological space, what would be an example of a retract of X which is not closed?

    I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

    [A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].
    Let T be a finite complement topology on Z_+. We know that 2Z_+ is not closed in (Z_+, T) (2 Z_+ is not finite in Z_+).

    Let r be a function r:Z_+ \rightarrow 2Z_+ such that r(2i-1)=r(2i)=2i for each i \in Z_+. r is well-defined. For any closed subset A of 2Z_+, r^{-1}(A) is closed in (Z_+, T). Thus r is continuous. We also have r(2i)=2i for each i in Z_+.

    Thus, 2 Z_+ is a retract of Z_+, which is not closed in (Z_+, T).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. A is a subset of a topological space X
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: August 28th 2011, 01:40 PM
  2. normal topological space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 5th 2010, 07:07 PM
  3. Topological space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 2nd 2010, 08:23 PM
  4. a topological space
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: March 8th 2009, 03:39 AM
  5. Proof in a topological space
    Posted in the Advanced Math Topics Forum
    Replies: 22
    Last Post: January 24th 2006, 07:50 AM

Search Tags


/mathhelpforum @mathhelpforum