Exercise 23, § 4

Let $p$ be a prime number, let $(G_n)$ be an infinite sequence of topological groups each identical with the discrete group $\mathbf{Z}/(p^2)$, and let $H_n$ be the subgroup $(p)/(p^2)$ of $G_n$. Let $G$ be the local direct product of the $G_n$ relative to the $H_n$ (§ 2, Exercise 26); then $G$ is a locally compact group but is not compact. Show that the continuous homomorphism $u : x \to px$ of $G$ into itself is not a strict morphism of $G$ onto $u(G)$, and that $u(G)$ is not closed in $G$.