Restricted product explained

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

Let

I be an index set; S a finite subset of I . If G _{i} is a locally compact group for each i \in I , and K _{i} \subset G _{i} is an open compact subgroup for each i \in I \setminus S , then the restricted product \prod _{i\nolimits'} G _{i} is the subset of the product of the G _{i} 's consisting of all elements (g _{i)} _{i} such that g _{i} \in K _{i} for all but finitely many i \in I \setminus S .

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

\prod _{i} A _{i,} where A _{i} is open in G _{i} and A _{i} = K _{i} 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.

See also

• Direct sum