Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 2G / Solution

The Baire category theorem says that a complete metric space cannot be a countable union of nowhere-dense closed sets. Equivalently, every countable intersection of open dense sets is dense.

A point $x$ of a metric space is an isolated point when some $\epsilon >0$ satisfies $B(x,\epsilon )=\{x\}$.