Corollary

Let $(G_i)_{i \in I}$ be a family of topological groups, and for each $i \in I$ let $H_i$ be a normal subgroup of $G_i$; let $\varphi_i$ denote the canonical mapping of $G_i$ onto $G_i/H_i$. Let $G = \prod_{i \in I} G_i, \ H = \prod_{i \in I} H_i$. Then the bijective homomorphism of $G/H$ onto $\prod_{i \in I} (G_i/H_i)$ associated with the continuous homomorphism $(x_i) \to (\varphi_i(x_i))$ is an isomorphism of topological groups.

For this homomorphism is an isomorphism of group structures.