Let $U$ be a simply connected open set in $\mathbb{R}^2$. Is it true that $U$ is homeomorphic to an open ball?
Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball?
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general-topology
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4Yes. This follows from the Riemann mapping theorem. – 2011-08-02
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0I'm asking more general question. – 2011-08-02
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0You have changed the question after correct answers have been posted. I think it'd be better to ask a separate question for general $n$. – 2011-08-02
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0Since you now re-asked your follow-up question I rolled back to the previous version. – 2011-08-02
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3BUT, isn't the fact in the question **MUCH EASIER TO PROVE** than the Riemann mapping theorem? (A snipe: is the empty set simply connected?) – 2011-08-02
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3@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
3 Answers
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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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3$\mathbb{R}^2$ is also homeomorphic to an open ball. – 2011-08-02
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1That's because the asker changed the question... – 2011-08-02