For my Topology final yesterday, I proved that the intersection of two open dense sets is again a dense set. I used the fact that a set $A$ being dense in a space $X$ is equivalent to $X - A$ having empty interior.
My proof, however, only seemed to use the fact that one of the two sets was open. My question is, can we get away with this? Given an open dense set $U_1$ and a dense set $U_2$, is $U_1 \cap U_2$ necessarily dense? The question is obviously not true if neither set is open, but I cannot think of a counterexample when only one set is open.