Baire category theorem

Baire category theorem. Let $(X,d)$ be a nonempty complete metric space. If $(U_n)_{n\in\mathbb{N}}$ is a sequence of dense open sets in $X$ , then $\bigcap_{n\in\mathbb{N}}U_n$ is dense in $X$ .

Equivalent characterizations

The dense-intersection conclusion is equivalent to saying that no nonempty open subset of $X$ is meager in $X$ . Consequently, $X$ is not a countable union of nowhere dense sets, and every residual set is dense. Thus every complete metric space is a Baire space.