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 $i\in I$, and $K_i\subset G_i$ is an open compact subgroup for each $i\in I\setminus S$, then the restricted product

$\prod _i{\mathrm{\backslash nolimits}}^{\prime }{G}_i$

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 $i\in I\setminus S$. 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). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.