# Prove or disprove: an open and simply connected subset of $\mathbb{R^2}$ is homeomorphic to the disk

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?

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.

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