I'm reading page 10, exercise 3 part (c) of:
http://www.mit.edu/~ssam/alggeom-I.pdf
in order to claim that if f is zero on a dense subset of $Y$ then f is zero everywhere don't we need that the codomain, i.e $k$ is a Hausdorff space? or we don't? in this case $k$ is an algebraically closed field and $f: Y \rightarrow k$ is a continuous map where $k$ has the Zariski topology so $k$ is non-Hausdorff.