Restricted product

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 $iin\; I$, $G\_i$ is a locally compact group, and for each , $K\_i\; subset\; G\_i$ is an open compact subgroup, then the restricted product : $\{prod\_i\}\text{'}\; G\_i,$is the subset of the product of the $G\_i$'s consisting of all elements $(g\_i)\_\{iin\; I\}$ such that $g\_i\; in\; K\_i$ for all but finitely many .

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.