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 $i\; \backslash in\; I$, and $K\_i\; \backslash subset\; G\_i$ is an open compact subgroup for each $i\; \backslash in\; I\; \backslash setminus\; S$, then the restricted product

: $\backslash prod\_i\backslash nolimits\text{'}\; G\_i\backslash ,$

is the subset of the product of the $G\_i$'s consisting of all elements $(g\_i)\_\{i\; \backslash in\; I\}$ such that $g\_i\; \backslash in\; K\_i$ for all but finitely many $i\; \backslash in\; I\; \backslash setminus\; S$.

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

: $\backslash prod\_i\; A\_i\backslash ,,$

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.