# Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball?

Let $U$ be a simply connected open set in $\mathbb{R}^2$. Is it true that $U$ is homeomorphic to an open ball?

• Yes. This follows from the Riemann mapping theorem.
– Mark
Aug 2, 2011 at 11:27
• I'm asking more general question. Aug 2, 2011 at 11:45
• You have changed the question after correct answers have been posted. I think it'd be better to ask a separate question for general $n$.
– lhf
Aug 2, 2011 at 11:47
• BUT, isn't the fact in the question MUCH EASIER TO PROVE than the Riemann mapping theorem? (A snipe: is the empty set simply connected?) Aug 2, 2011 at 14:39
• @GEdgar: Proving this without the Riemann mapping theorem was the subject of this MathOverflow question. Aug 2, 2011 at 16:26

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.
According to the Riemann mapping theorem that's true iff U is a simply connected nonempty open set in $\mathbb{R}^2$ which is a strict subset. That is, $U\subsetneq \mathbb{R}^2$.
• $\mathbb{R}^2$ is also homeomorphic to an open ball. Aug 2, 2011 at 11:37