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).

Proof sketch (optional): To show density, fix a nonempty open ball $B_1$. Since $U_1$ is dense, pick a smaller ball $B_2\subseteq B_1\cap U_1$. Continue inductively to get nested balls $B_{n+1}\subseteq B_n\cap U_n$ with radii $\to 0$. Completeness ensures the intersection contains a point lying in every $U_n$ and hence in the desired intersection.