In a Hausdorff topological space, $(X, T)$, with non-empty subset $A \subset X$, how can we prove that an open set $U \in T$ has non-empty intersection with $A$ iff $U$ has non-empty intersection with $\overline{A}$?
The $\Leftarrow$ direction is obvious, but I'd be interested in seeing how we can prove the opposite relation holds.