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

True. Part (v) supplies one isolated point $p$. Since $\{p\}$ is open, $X\setminus \{p\}$ is closed and therefore complete. It is nonempty when $X$ has at least two points, so part (v) applied to this subspace supplies an isolated point $q$. Because $d(p,q)>0$, a sufficiently small ball showing that $q$ is isolated in the subspace also avoids $p$; hence $q$ is isolated in $X$.