Restricted product

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

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

∏ i ′ G i {\displaystyle \prod _{i}\nolimits 'G_{i}\,}

is the subset of the product of the  G i {\displaystyle G_{i}} 's consisting of all elements  ( g i ) i ∈ I {\displaystyle (g_{i})_{i\in I}} such that  g i ∈ K i {\displaystyle g_{i}\in K_{i}} for all but finitely many  i ∈ I ∖ S {\displaystyle i\in I\setminus S} .

This group is given the topology whose basis of open sets are those of the form

∏ i A i , {\displaystyle \prod _{i}A_{i}\,,}

where  A i {\displaystyle A_{i}} is open in  G i {\displaystyle G_{i}} and  A i = K i {\displaystyle A_{i}=K_{i}} for all but finitely many  i {\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

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* Direct sum

## References

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* 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">