Show that this topology is Hausdorff

Question

Let $(\widetilde{D}, \pi)$ be a Riemann domain over $\mathbb{C}^n$. We introduce a topology on $\widetilde{D}$ in the following manner. A neighbourhood of a point $Z = (z, f_z^{\alpha}) \in \widetilde{D}$ will be defined to be the set of all points $W = (w, f_w^{\beta}) \in \widetilde{D}$ such that

1. For a fixed $\epsilon >0$, $\| z-w \| <\epsilon$
2. The germ $f_w^{\beta}$ is represented by an element $(V_w, f^{\alpha}) \in f_z^{\alpha}$ which is a direct analytic continuation of $(U_z, f^{\alpha}) \in f_z^{\alpha}$.

I am comfortable with basic topology proofs. My concern here is in condition 2. My proof is a little ''hand wavy" and I would like a rigorous proof.