Restricted product

In mathematics, 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.

See also

• Direct sum

References

• Fröhlich, A.; Cassels, J. W. (1967), Algebraic number theory, Boston, MA: Academic Press, ISBN 978-0-12-163251-9
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859