The closed-set version of the <Baire category theorem> says that if a nonempty <complete metric space> is a <countable union> of <closed sets>, at least one of those <closed sets> 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 / 2008 / ii / Paper 3 / 2F / a / Solution

 Solution (source code)