Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 2F / i / Solution

The closed-set version of the Baire category theorem says that if a nonempty complete metric space $X$ is a countable union $X={\bigcup }_{n=1}^{\infty }{F}_n$ of closed sets, then at least one $F_n$ has nonempty interior. Equivalently, a nonempty complete metric space cannot be a countable union of closed nowhere dense sets.