# homeomorphisms and interior, boundary

• Feb 24th 2009, 08:02 PM
Andreamet
homeomorphisms and interior, boundary
Show that if f: X->Y is a homeomorphism, then:

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

I am stuck!
• Feb 25th 2009, 02:28 AM
aliceinwonderland
Quote:

Originally Posted by Andreamet
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.