TITLE: Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball?
QUESTION [14 upvotes]: Let $U$ be a simply connected open set in $\mathbb{R}^2$. Is it true that $U$ is homeomorphic to an open ball?

REPLY [10 votes]: Yes, this is the Riemann mapping theorem. You get much more than a homeomorphism: you get a biholomorphic map.

REPLY [2 votes]: Yes.  In fact, more can be said... The Riemann Mapping Theorem states that the homeomorphism can be taken to be biholomorphic (as a complex map), if $U \neq \mathbb{C}$.  See this link for a much more detailed treatment and proof.
Hope this helps!