# Reference request: 2-dimensional Schonflies theorem

Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ extends to a homeomorphism $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. The discussions of the Jordan Curve Theorem that I can remember don't prove this stronger statement.

This statement is mentioned on the Wikipedia page for the Schoenflies problem . I looked through several papers on the generalized Schoenflies problem (which requires extra hypotheses in higher dimensions to rule out things like the Alexander Horned Sphere), but no luck...

• Check out Moise's book: ams.org/mathscinet-getitem?mr=488059 – Ian Agol May 24 '12 at 23:01
• See Berenstein-Gay "complex variables". Krantz and Bell have several papers on this as well. – Andrés E. Caicedo May 24 '12 at 23:04
• Another possible source is Kai-Uwe Bux's "Notes on Geometric Topology" which has a section on the Schoenflies theorem. This can be found on his webpage at Bielefeld. – Allen Hatcher May 25 '12 at 3:12
• I thought it was in Bing's book. – Scott Carter May 25 '12 at 6:42