Prove that if is a continuous map of a space X to a Hausdorff space Y then its graph
is a closed subset of X x Y.
Let (x,y) be a point in the closure of , and let be a sequence of points in with limit point (x,y). Then and . But f is continuous, and therefore . Therefore y=f(x) and hence . Thus every limit point of is in , and so is closed.
(If the space is "too big" for closures to be detected by sequences then you will have to replace "sequence" by "directed net" in that proof.)
The proof does use the information that Y is Hausdorff (though I didn't actually point that out). It is used to deduce that if and then . (In a Hausdorff space, a sequence has a unique limit. In a non-Hausdorff space, it's possible for a sequence to have more than one limit.)