Category Argument Template

Category argument template: Let $X$ be a Baire space and let $U_1,U_2,\dots \subseteq X$ be sets that are each open and dense in $X$. Then the intersection $\bigcap_{n=1}^\infty U_n$ is dense (in particular, nonempty) in $X$, and it is a residual set since its complement is a countable union of nowhere dense sets .

Conceptually, this packages the Baire category theorem into a reusable method: encode the $n$th requirement of a property as membership in a dense open set $U_n$, and then conclude that there are points satisfying all requirements at once (finite versions rely only on intersection of dense open sets being dense ).