Exercise 28, § 5

Let $G, G'$ be two topological groups. Show that every continuous homomorphism $f$ of $G$ onto $G'$ is a strict morphism in each of the following two cases:

a) $G$ is locally compact and $\sigma$-compact, and $G'$ is non-meagre [show that there is a relatively compact open set $U$ in $G$ such that the interior of $f(\overline{U})$ is not empty; deduce that for each compact neighbourhood $V$ of the identity element of $G$, the interior of $f(V)$ is not empty, and then apply Chapter III, § 2, Exercise 18].

b) $G$ is complete, the topology of $G$ has a countable base, and $G'$ is non-meagre. [By using the fact that, for each neighbourhood $U$ of the identity element $e$ of $G$, $G$ is the union of a sequence of sets of the form $x_n U$, show that for each open set $A$ in $G$ there is an open set $A'$ in $G'$ which contains $f(A)$ and is such that $f(A)$ is dense in $A'$. Then consider a fundamental system $(U_n)$ of symmetric neighbourhoods of $e$ in $G$, such that $\overline{U}_{n+1}^2 \subset U_n$, and show that $f(U_p)$ contains the open set $U'_{p+1}$; to do this, take any point $a' \in U'_{p+1}$ and then construct by induction a sequence $(b_n)$ of points of $G$ such that $b_n \in U_{p+n}$ and such that $f(b_1 b_2 \ldots b_n)$ tends to $a'$; note that, if $x' \in U'_k$, then there exists $y \in U_k$ such that $f(y) \in x'U_{k+1}'$.] The hypothesis that $G$ has a countable base is essential, as is shown by the example where $G'$ is the group $\mathbf{R}$ with the topology of the real line, $G$ is the group $\mathbf{R}$ with the discrete topology and $f : G \to G'$ is the identity mapping.