Restricted product

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 \( {\displaystyle K_{i}\subset G_{i}} \) is an open compact subgroup for each \( {\displaystyle i\in I\setminus S} \), then the restricted product

\( {\displaystyle {\prod _{i}}'G_{i}\,} \)

is the subset of the product of the \( {\displaystyle G_{i}} \) 's consisting of all elements \( {\displaystyle (g_{i})_{i\in I}} \) such that \( {\displaystyle g_{i}\in K_{i}} \) for all but finitely many \( {\displaystyle i\in I\setminus S}. \)

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

\( {\displaystyle \prod _{i}A_{i}\,,} \)

where \( A_{i} \) is open in \( G_i \) and \( {\displaystyle A_{i}=K_{i}} \) 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.