Biholomorphism between an open set and $\mathbb C^n$

Question

If $U$ is a polydisc in $\mathbb C^n$, that is, $U=\{z \in \mathbb C^n:|z_i|<1\}$, can we find a biholomorphic map from $U$ to $\mathbb C^n$?

Answer 1

In an open set biholomorphic to $\mathbb C^n$ there are no bounded holomorphic functions which are not constant.

Answer 2

In the case $n = 1$, the unit disc is not biholomophic to $\mathbb C$.