restricted direct product

Let {Gv}vV be a collection of locally compact topological groups. For all but finitely many vV, let HvGv be a compact open subgroup of Gv. The restricted direct productPlanetmathPlanetmathPlanetmath of the collection {Gv} with respect to the collection {Hv} is the subgroupMathworldPlanetmathPlanetmath

G:={(gv)vVvVGv|gvHv for all but finitely many vV}

of the direct productPlanetmathPlanetmathPlanetmathPlanetmath vVGv.

We define a topology on G as follows. For every finite subset SV that contains all the elements v for which Hv is undefined, form the topological groupMathworldPlanetmath


consisting of the direct product of the Gv’s, for vS, and the Hv’s, for vS. The topological group GS is a subset of G for each such S, and we take for a topology on G the weakest topology such that the GS are open subsets of G, with the subspace topology on each GS equal to the topology that GS 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