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

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

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.Grundlehren der mathematischen WissenschaftenCategory:Articles containing German-language text&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https://zbmath.org/?format=complete&q=an:0956.11021#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1697859#id-name=MR&rft.isbn=978-3-540-65399-8&rft.aulast=Neukirch&rft.aufirst=Jürgen&rfr_id=info:sid/en.wikipedia.org:Restricted product" class="Z3988">