PlanetMath: restricted direct product

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restricted direct product

(Definition)

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.

"restricted direct product" is owned by djao.

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Cross-references: right, subspace topology, open subsets, contains, subset, finite, topology, direct product, subgroup, open subgroup, compact, topological groups, locally compact, collection
There are 4 references to this entry.

This is version 2 of restricted direct product, born on 2002-04-17, modified 2003-12-31.
Object id is 2845, canonical name is RestrictedDirectProduct.
Accessed 1867 times total.

Classification:

AMS MSC:	11R56 (Number theory :: Algebraic number theory: global fields :: Adèle rings and groups)
	22D05 (Topological groups, Lie groups :: Locally compact groups and their algebras :: General properties and structure of locally compact groups)

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