Is every Zariski dense set dense with regard to the standard topology?

Question

I was wondering whether this is true or not, since it could turn in a powerful tool when working on the standard topology.

Answer 1

No. For example, every infinite subset of the affine line over $\mathbb{C}$ is Zariski dense, but there are plenty of infinite subsets of $\mathbb{C}$ which are not dense in the standard topology.

The converse, however, is true, since the Zariski topology is coarser than the standard topology.