1. ## Topology Proof

I need some help with a topological proof, hopefully there is someone out there that can help.

Prove that a bijection f: X→Y is a homeomorphism if and only if f and f^-1 map closed sets to closed sets.

Thanks

2. The definition of a homeomorphism is a map which is bijective continuous and open (maps open sets to open sets).

Assume $f: X \rightarrow Y$ is a homeomorphism.

$f$ is continuous and open so $f^{-1}$ is continuous and open. But then $f$ is closed (maps closed sets to closed sets) and so is $f^{-1}$ since let $C$ be a closed subset of $Y$. Then $C^c$ is open in $Y$ and so $f^{-1}(C^c)$ is open in $X$. But $f^{-1}(C^c)=(f^{-1}(C))^c$ and therefore $f^{-1}(C) = (f^{-1}(C)^c)^c$ is closed in $X$. Same argument shows $f$ is a closed map.

Now assume that $f$ and $f^{-1}$ are bijective maps which take closed sets to closed sets. We need to show that $f$ is continuous and open. Consider an arbitrary open set $U \subset Y$. $U^c$ is closed and $f^{-1}(U^c) = f^{-1}(U)^c$ is closed in $X$. Therefore $f^{-1}(U)$ is open in $X$. So $f$ is continuous. Similar argument shows $f$ is open.

3. THanks, but i have one question are C^c and U^c the compliments of C and U? Just want to make sure.

Thanks

4. Originally Posted by reagan3nc
THanks, but i have one question are C^c and U^c the compliments of C and U? Just want to make sure.
Yes.