Riemann mapping theorem

Statement

Any simply connected domain in $\mathbb {C}$ (i.e. any open, connected and simply connected subset of $\mathbb {C}$) that is not the whole of $\mathbb {C}$, is bihomolorphically equivalent to the open unit disc.

Corollaries

Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to $\mathbb {C}$).