Definition 2

The topological group obtained by giving the product set

the group structure which is the product of the group structures of the $G_i$ and the topology which is the product of the topologies of the $G_i$ is called the product of the topological groups $G_i$.

If $(J_x)_{x \in K}$ is a partition of $I$, then $G$ is isomorphic to the product of the topological groups $\prod_{i \in J_x} G_i$ (associativity of the product).

If $H_i$ is a subgroup of $G_i$, then the product of the topological groups $H_i$ is isomorphic to the subgroup $\prod_{i \in J} H_i$ of $\prod_{i \in I} G_i$. In particular, if $J$ is any subset of $I$, and $J' = C J$, then the topological group $\prod_{i \in J} G_i$ is isomorphic to the normal subgroup $G'_J = (\prod_{i \in J} G_i) \times (\prod_{i \in J'} \{e_i\})$ of $G$. Since the projection of every open set is an open set, the projection $\mathrm{pr}_J$ of $G$ onto $\prod_{i \in J} G_i$ is a strict morphism, and consequently the quotient group $G/G'$ is isomorphic to $G'_J$; $G$ is isomorphic to the product $G'_J \times (G/G'_J)$.