Corollary 2

Let $(G_\alpha, f_{\alpha \beta})$ be an inverse system of topological groups which satisfies the conditions of Proposition 5, and let $G'$ be a topological group in which there is a neighbourhood $V'$ of the identity element $e'$ which contains no subgroup of $G'$ other than $\{ e' \}$. Then if $v : G \to G'$ is any continuous homomorphism, there is an index $\alpha \in I$ and a continuous homomorphism $v_\alpha : G_\alpha \to G'$ such that $v = v_\alpha \circ f_\alpha$.

For since $\overline{v}^{-1}(V')$ is a neighbourhood of $e$ in $G$, there is an index $\alpha$ and a neighbourhood $V_\alpha$ of $e_\alpha$ in $G_\alpha$ such that $f_\alpha^{-1}(V_\alpha) \subset \overline{v}^{-1}(V')$. Hence $v(H_\alpha) \subset V'$, and since $v(H_\alpha)$ is a subgroup of $G'$, it follows that $v(H_\alpha) = \{ e' \}$. Since $f_\alpha$ may be identified with the canonical mapping $G \to G/H_\alpha$, the corollary is a consequence of the canonical factorization of a continuous homomorphism (§ 2, no. 8).