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 $i\in I$, G_i is a locally compact group, and for each $i\in I\backslash S$, $K_i \subset G_i$ is an open compact subgroup, 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 $i\in I\backslash S$.

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.

References

• Fröhlich, A.; Cassels, J. W. (1967), Algebraic number theory, Boston, MA: Academic Press, ISBN 978-0-12-163251-9