Restricted product

In mathematics, the restricted product is a construction in the theory of topological groups.

be an index set;

a finite subset of

is a locally compact group for each

is an open compact subgroup for each

, then the restricted product is the subset of the product of the

's consisting of all elements

for all but finitely many

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

for all but finitely many

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.