Analytic germ projection ($\pi ([f{]}_z)=z$)

The projection $[f{]}_z\mapsto z$ is a local biholomorphism for the sheet topology on the space of germs of holomorphic functions. It need not be a topological covering map: branches may end at singularities even when another branch remains over that base point.

The inverse germs of $w^3-3w$ over $\mathbb{C}$ form the connected surface $\mathbb{C}\setminus \{-1,1\}$. Its projection is surjective and locally biholomorphic, but the fibres have three points generically and one point at each critical value $\pm 2$, so it is not an evenly covered topological covering.