Riemann domain over C^n

Remarks

The charts $\pi|_V$ make $X$ a complex manifold of dimension $n$ on which $\pi$ is locally biholomorphic. A schlicht Riemann domain is the same as a domain in $\C^n$, through the identification of $X$ with $\pi(X)$.

Example: $X = \C^2$ with $\pi(w, z_2) = (e^w, z_2)$. The space $X$ is connected and Hausdorff, and $\pi$ is a local homeomorphism because $w \mapsto e^w$ is a local homeomorphism of $\C$ onto $\C \setminus \set{0}$. It is not schlicht, since $\pi(w + 2\pi i, z_2) = \pi(w, z_2)$.

Non-example: $X = \C^2$ with $\pi(w, z_2) = (w^2, z_2)$. Here $\pi$ is not injective on any neighbourhood of a point $(0, z_2)$, because $(w, z_2)$ and $(-w, z_2)$ have the same image; so $\pi$ is not a local homeomorphism there.