Exercise 22, § 2

Let $G$ be the product of a family $(G_i)_{i \in I}$ of topological groups, and let $G'$ be a topological group such that there is a neighbourhood $V'$ of the identity element $e'$ in $G'$ which contains no normal subgroup of $G'$ other than $\{e'\}$. If $f$ is a continuous homomorphism of $G$ into $G'$, show that there is a subset $J$ of $I$ whose complement $J'$ is finite, such that $f$ is equal to $e'$ on the subgroup $\prod_{i \in I} G_i \times \prod_{i \in J'} \{e_i\}$ of $G$.