Let $(X, d)$ be a [complete metric space](/page/Complete%20Metric%20Space). Then $X$ is a Baire space. That is:

1. **Dense $G_\delta$ form.** If $(G_n)_{n=1}^\infty$ is a sequence of dense open subsets of $X$, then $\bigcap_{n=1}^\infty G_n$ is dense in $X$. 2. **Category form.** If $(F_n)_{n=1}^\infty$ is a sequence of [nowhere dense](/page/Nowhere%20Dense%20Set) closed subsets of $X$, then $\bigcup_{n=1}^\infty F_n$ has empty interior. In particular, $X \neq \bigcup_{n=1}^\infty F_n$ whenever $X$ is nonempty.