Exercise 12, § 3

a) Let $G'$ be a complete Hausdorff group; let $G_0$ be a dense subgroup of $G'$, distinct from $G'$; and let G be the topological group obtained by giving $G_0$ the discrete topology. The identity mapping $G \to G_0$ is then a bijective continuous homomorphism, but its continuous extension $\hat{G} \to \hat{G}_0$ is not surjective.

\* b) Let G be the group $\mathbf{Q}^2$, $\theta$ an irrational number, u the continuous homomorphism $(x, y) \to x + \theta y$ of G into $\mathbf{R}$, and let $G' = u(G)$. Then $u : G \to G'$ is bijective, but its continuous extension $\hat{G} \to \hat{G}'$ is not injective.

c) From a) and b) construct an example of a bijective continuous homomorphism $G \to G'$ of Hausdorff commutative groups whose continuous extension $\hat{G} \to \hat{G}'$ is neither injective nor surjective *.