Prove that that a map in $C^N$ in the metric topology is continuous to the Zariski topology, but that the map is not a homeomorphism.
My Work:
I plan to use the fact that the metric topology is a Hausdorff space, whereas the Zariski topology in $C^N$ is not a Hausdorff, but then wouldn't the map not be continuous?
If I interpret your question, you want to show that the identity map of $(C^n,d)\rightarrow (C^n,Zar)$ is continuous, which also means that the Zariski topology is weaker than the metric topology. This is a consequence of the fact that a Zariski open subset is also open for the metric topology.
On the other hand $Id:(C^n,Zar)\rightarrow (C^n,d)$ is not continuous, since there exists open subsets of $(C^n,d)$ like the open ball $B(0,1)$ which are not Zariski open, so their inverse image by the identity is not Zariski open.