I want to prove that the topology on $\bigsqcup \limits_{z \in \mathbb{C}^n} \mathcal{O}_z$, where the stalks are germs of holomorphic functions is Hausdorff.
I distinguish two cases:
$x \neq y$ lie in different fibers of $\bigsqcup \limits_{z \in \mathbb{C}^n} \mathcal{O}_z$. I was able to seperate them without even using that we are dealing with holomorphic functions.
$x,y \in \mathcal{O}_{z_0}$ for some $z_0 \in \mathbb{C}^{n}$. Now I can find holomorphic functions $f,g$ on an open set $U \subset \mathbb{C}^n$ containing $z_0$ such that $f_{z_0} = x$, $g_{z_0} = y$. In particular, $f$ and $g$ must be different on every open set $z_0 \in V \subset U$, otherwise they would be equivalent.
I was told in the proof the Identity Theorem must be used. Probably in its contrapositive form. But I don't see how.