Baire category theorem

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

Equivalently, $X$ is not a countable union of nowhere dense sets ; in particular, no nonempty open set is meager , and every residual set is dense. This theorem is commonly summarized by saying that complete metric spaces are Baire spaces .