Intersection of Dense Open Sets is Dense

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $X$ .

Explanation

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).