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Intuitively it seems that if a set $S \subset \mathbb{R^2} $ is open and simply connected it must be homeomorphic to the open (unit) disk.

I think the same would hold for a closed and bounded simply connected subset, it would be homeomorphic to the closed disk.

Is this true? How could you prove it, or what would be a counterexample?

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Is what true? The statement about open sets is true; this follows from the Riemann mapping theorem. The statement about closed sets is false; for example consider a closed line segment.

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  • $\begingroup$ Is it true that this is the two-dimensional Poincaré conjecture? $\endgroup$
    – Parcly Taxel
    Dec 5, 2017 at 17:03
  • $\begingroup$ Yes, the RMT plus the map $z\to z/(1+|z|)$ on the entire plane. $\endgroup$
    – zhw.
    Dec 5, 2017 at 17:55
  • $\begingroup$ @zhw. Well of course - the case of the entire plane is why I said "follows from" instead of something stonger... $\endgroup$
    – David C. Ullrich
    Dec 5, 2017 at 18:01

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