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>.

Solution

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

 Solution (source code)