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

${\prod _{i}}'G_{i}\,$

is the subset of the product of the $G_{i}$ 's consisting of all elements $(g_{i})_{i\in I}$ such that $g_{i}\in K_{i}$ 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

$\prod _{i}A_{i}\,,$

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.