The closed-set form of the <Baire category theorem> says: if a nonempty <complete metric space> is a <countable union> of <closed sets>, at least one of those sets has nonempty <interior>. Equivalently, a complete metric space cannot be a <countable union> of <closed sets> with empty interiors. In the application the space is $\mathbb R$ with its usual complete metric.

Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 2G / a / Solution

 Solution (source code)