Baire Category Theorem

Baire Category Theorem: If $(X,d)$ is a complete metric space and $U_1,U_2,\dots$ are open dense subsets of $X$, then $\bigcap_{n=1}^\infty U_n$ is dense in $X$. Equivalently, $X$ is not a countable union of nowhere dense sets .

Baire’s theorem is a powerful structural result with many analytic consequences (existence of “typical” points, uniform boundedness phenomena, and more).