Restricted product

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The restricted product is a construction in the theory of topological groups.

Let I be an indexing set; S a finite subset of I. If for each i\in I, Gi is a locally compact group, and for each i\in I\backslash S, K_i \subset G_i is an open compact subgroup, then the restricted product

{\prod_i}' G_i\,

is the subset of the product of the Gi's consisting of all elements (g_i)_{i\in I} such that g_i \in K_i for all but finitely many i\in I\backslash S.

This group is given the topology whose basis of open sets are those of the form

\prod_i A_i\,,

where Ai is open in Gi and Ai = Ki for all but finitely many i.

One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.

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