Let $\{G_{v}\}_{{v\in V}}$ be a collection of locally compact topological groups. For all but finitely many $v\in V$, let $H_{v}\subset G_{v}$ be a compact open subgroup of $G_{v}$. The restricted direct product of the collection $\{G_{v}\}$ with respect to the collection $\{H_{v}\}$ is the subgroup

of the direct product $\prod_{{v\in V}}G_{v}$.

We define a topology on $G$ as follows. For every finite subset $S\subset V$ that contains all the elements $v$ for which $H_{v}$ is undefined, form the topological group

consisting of the direct product of the $G_{v}$’s, for $v\in S$, and the $H_{v}$’s, for $v\notin S$. The topological group $G_{S}$ is a subset of $G$ for each such $S$, and we take for a topology on $G$ the weakest topology such that the $G_{S}$ are open subsets of $G$, with the subspace topology on each $G_{S}$ equal to the topology that $G_{S}$ already has in its own right.