homeomorphisms and interior, boundary

**Andreamet** · February 24th 2009, 07:02 PM

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

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

I am stuck!

**aliceinwonderland** · February 25th 2009, 01:28 AM

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.

Thread: homeomorphisms and interior, boundary

LinkBack

Thread Tools

Display

homeomorphisms and interior, boundary

Similar Math Help Forum Discussions

Interior, Closure and Boundary of sets

identify interior points, boundary points, ......

Need help understanding interior points, boundary points, isolated points, etc.

The interior of a set is open and the boundary and closure of a set is closed

Interior, Boundary, and Closure of Sets

Search Tags