Complex plane and unit disc are homeomorphic but not conformally equivalent
= Complex plane and unit disc are homeomorphic but not conformally equivalent
The radial map $z\mapsto z/(1+|z|)$ is a <homeomorphism> from $\mathbb C$ to the unit disc, with inverse $w\mapsto w/(1-|w|)$. A conformal equivalence in that direction would be a nonconstant bounded entire function, contradicting the <Liouville theorem>.