Q on Riemann domains

lark

If P is the projection map from a Riemann domain M [itex]\rightarrow C^n[/itex], and U is a connected subset of M with P(U)=B, where B is a ball in [itex]C^n[/itex], then is P injective on U, so it's a homeomorphism on U?
P is locally a homeomorphism by definition.
It would be related to B being simply connected. WOuldn't be true if P(U) were ring-shaped.
I read something saying that in complex analysis, local homeomorphisms being global homeomorphisms relates to connectivity.
If P is proper, meaning if K is a compact subset of B, the inverse image in U of K is compact, it would be true by a theorem of Ho, apparently.
Laura