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Thread: homeomorphisms and interior, boundary

  1. #1
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    homeomorphisms and interior, boundary

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

    $\displaystyle f(\partial(A))=\partial(f(A))$

    I am stuck!
    Last edited by Andreamet; Feb 24th 2009 at 07:59 PM.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Show that if f: X->Y is a homeomorphism, then:

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

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

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

    Now, it remains to combine 1 & 2 to get the answer.
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