Exercise 7, § 2

Let $X$ be an infinite discrete space. Show that the uniformity of finite partitions (Chapter II, § 2, no. 2) on $X$, which is compatible with the topology of $X$, is not metrizable (otherwise there exists a sequence $(\mathfrak{F}_n)$ of finite partitions of $X$ such that every finite partition of $X$ is formed of sets, each of which is a union of sets belonging to one of the partitions $\mathfrak{F}_n$; from this it would follow that the set of finite partitions of $X$ is countable).