Baire category theorem

Statement

Each complete metric space is a Baire space.

Remarks

Explicitly: if $X$ is a complete metric space and $(U_n)_{n \in \mathbb{N}}$ is a sequence of open dense subsets of $X$, then $\bigcap_n U_n$ is dense in $X$. The proof builds a nested sequence of closed balls with radii tending to $0$, and uses completeness to find a point in all of them. A second form of the theorem says that each locally compact Hausdorff space is a Baire space. A consequence: $\mathbb{R}$ is not a countable union of nowhere dense sets, and thus $\mathbb{R}$ is uncountable.

Explanations

L1 human

A complete metric space cannot be covered by countably many thin sets. For example, you cannot write the real line as a countable union of sets like $\mathbb{Z}$ or the Cantor set.