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\{\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
Canonical name	RestrictedDirectProduct
Date of creation	2013-03-22 12:35:38
Last modified on	2013-03-22 12:35:38
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Definition
Classification	msc 11R56
Classification	msc 22D05