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

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    @GEdgar: Proving this without the Riemann mapping theorem was the subject of [this MathOverflow question](http://mathoverflow.net/questions/66048/riemann-mapping-theorem-for-homeomorphisms).2011-08-02

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

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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!

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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, $U\subsetneq \mathbb{R}^2$.

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    Tha$t$'s because the aske$r$ chan$g$ed t$h$e que$s$tion...2011-08-02