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Compact Collection Open Set Set Finite Superset Topological Space Subspace Topology Convex Angle Subgroup Direct Product And Restricted Direct Product Of Groups Topological Group Locally Compact Groupoids Full Subgroup

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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 . The restricted direct product of the collection $\{G_v\}$ with respect to the collection $\{H_v\}$ is the subgroup

$\displaystyle G := \left\{ \left. (g_v)_{v \in V} \in \prod_{v \in V} G_v\ \right\vert \ 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 as follows. For every finite subset $S \subset V$ that contains all the elements for which is undefined, form the topological group

$\displaystyle G_S := \prod_{v \in S} G_v \times \prod_{v \notin S} H_v$

consisting of the direct product of the

's, for $v \in S$ , and the

's, for $v \notin S$ . The topological group

is a subset of

for each such

, and we take for a topology on

the weakest topology such that the

are open subsets of

, with the subspace topology on each

equal to the topology that

already has in its own right.

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