We know by Riemann mapping theorem that for every connected, simply connected proper open subset $U$ of the complex plane, there exists a biholomorphic map $\phi:U\to \mathbb{D}$ (Where by $\mathbb{D}$ we mean the unit open disk). The proof of this theorem is too complicated.
Is there a simple proof for the similar question, when we are looking for homeomorphism or diffeomorphism instead of holomrphic function?
