In mathematics, the restricted product is a construction in the theory of topological groups.
Let be an index set; a finite subset of . If is a locally compact group for each , and is an open compact subgroup for each , then the restricted product
is the subset of the product of the 's consisting of all elements such that for all but finitely many .
This group is given the topology whose basis of open sets are those of the form
where is open in and for all but finitely many .
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
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.
