Intersection of Dense Open Sets is Dense

Intersection of dense open sets is dense: Let $X$ be a topological space and let $U,V\subseteq X$. If $U$ and $V$ are open and dense in $X$, then $U\cap V$ is open and dense in $X$.

This uses that a topology is closed under finite intersections and is a basic ingredient in many Baire category arguments (where one wants to intersect many dense open conditions).