# 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 of the collection $\{G_{v}\}$ with respect to the collection $\{H_{v}\}$ is the subgroup

 $G:=\left\{\left.(g_{v})_{v\in V}\in\prod_{v\in V}G_{v}\ \right|\ g_{v}\in H_{v% }\text{ for all but finitely many v\in V}\right\}$

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.

Title restricted direct product RestrictedDirectProduct 2013-03-22 12:35:38 2013-03-22 12:35:38 djao (24) djao (24) 5 djao (24) Definition msc 11R56 msc 22D05