Results 1 to 2 of 2

Math Help - homeomorphisms and interior, boundary

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    63

    homeomorphisms and interior, boundary

    Show that if f: X->Y is a homeomorphism, then:

    f(\partial(A))=\partial(f(A))

    I am stuck!
    Last edited by Andreamet; February 24th 2009 at 08:59 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by Andreamet View Post
    Show that if f: X->Y is a homeomorphism, then:

    f(\partial(A))=\partial(f(A))
    I assume A is a subset of X.

    Since  \partial A = \overline{A} \cap \overline {X \setminus A}, f (\partial A) = f( \overline{A} \cap \overline {X \setminus A}).
    We need to show that f( \overline{A} \cap \overline {X \setminus A}) is \overline{f(A)} \cap \overline {Y \setminus f(A)}, which is \partial (f(A)).

    1. For every subset A of X, one has f(\bar{A}) \subset \overline{f(A)} when f is continuous. If f is a homeomorphism, f(\bar{A}) = \overline{f(A)}.
    2. f(X \setminus A) = (Y \setminus f(A)). Using 1, f(\overline{X \setminus A}) = \overline{Y \setminus f(A)}.

    Now, it remains to combine 1 & 2 to get the answer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Interior, Closure and Boundary of sets
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 18th 2011, 12:48 PM
  2. identify interior points, boundary points, ......
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: July 24th 2011, 06:02 AM
  3. Replies: 1
    Last Post: February 9th 2011, 11:52 AM
  4. Replies: 5
    Last Post: September 4th 2010, 09:58 AM
  5. Interior, Boundary, and Closure of Sets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 22nd 2009, 05:01 PM

Search Tags


/mathhelpforum @mathhelpforum