Open mapping theorem for holomorphic functions

Given: A nonconstant holomorphic function $f$ on a complex domain $\mathrm{\Omega }$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$) and an open subset $O\subseteq \mathrm{\Omega }$.

If two holomorphic functions on a complex domain agree on a set with an accumulation point in the domain, then they agree everywhere on the domain (Identity theorem for holomorphic functions).

If $\mathrm{\Omega }$ is a complex domain, $f:\mathrm{\Omega }\to \mathbb{C}$ is nonconstant and holomorphic, $a\in \mathrm{\Omega }$, and $m={\mathrm{deg}}_{a}f$, then near $a$ there is a biholomorphic coordinate $\varphi$ with $\varphi (a)=0$ and $f(z)-f(a)=\varphi (z{)}^m$ (Local normal form of a nonconstant holomorphic map).

If $\mathrm{\Omega }$ is a complex domain, $f:\mathrm{\Omega }\to \mathbb{C}$ is nonconstant and holomorphic, $a\in \mathrm{\Omega }$, and $m={\mathrm{deg}}_{a}f$, then every neighbourhood $N$ of $a$ contains an open neighbourhood $V$ for which some $\rho >0$ gives exactly $m$ preimages in $V$ for $0<\mathrm{\mid }w-f(a)\mathrm{\mid }<{\rho }^m$, while $f(a)$ has only the preimage $a$, counted with multiplicity $m$ (A local degree-m holomorphic map has m nearby sheets).