Let $M$ and $N$ be complex manifolds and $f \colon M \rightarrow N$ be an injective holomorphic map. Is $f$ necessarily a topological embedding? In other words, is $f$ necessarily a homeomorphism from $M$ onto its image $f(M)$?
In particular, is every immersed submanifold of a complex manifold necessarily an embedded submanifold? An immersed submanifold of a complex manifold $N$ is the image of a complex manifold $M$ under an injective holomorphic immersion $f \colon M \rightarrow N$. An embedded submanifold of a complex manifold $N$ is the image of a complex manifold $M$ under an holomorphic embedding $f \colon M \rightarrow N$ (that is, a holomorphic immersion that is also a topological embedding).
In the context of smooth manifolds, it is well known that the analog questions have a negative answer. I believe that this is still false here, but I have not been able to find any counterexample.