Restricted product

In mathematics, the restricted product is a construction in the theory of topological groups.

Let ${\displaystyle I}$ be an index set; ${\displaystyle S}$ a finite subset of ${\displaystyle I}$. If ${\displaystyle G_{i}}$ is a locally compact group for each ${\displaystyle i\in I}$, and ${\displaystyle K_{i}\subset G_{i}}$ is an open compact subgroup for each ${\displaystyle i\in I\setminus S}$, then the restricted product

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

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

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

where ${\displaystyle A_{i}}$ is open in ${\displaystyle G_{i}}$ and ${\displaystyle A_{i}=K_{i}}$ for all but finitely many ${\displaystyle i}$.

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. MR 1697859. Zbl 0956.11021.