Restricted product

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

Let be an indexing set; a finite subset of . If for each , is a locally compact group, and for each , is an open compact subgroup, 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


  • 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.
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