restricted direct product

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

$$G:=\left\{{(g_v)}_{v\in V}\in \prod _{v\in V}{G}_v\right|g_v\in H_v\text{ for all but finitely many }v\in V\}$$

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

$$G_S:=\prod _{v\in S}{G}_v\times \prod _{v\notin S}{H}_v$$

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.