Restricted product

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

Let $I$ be an indexing set; $S$ a finite subset of $I$ . If for each $i\in I$ , $G_{i}$ is a locally compact group, and for each $i\in I\backslash S$ , $K_{i}\subset G_{i}$ is an open compact subgroup, then the restricted product

{\prod _{i}}'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 $i\in I\backslash S$ .

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.

References

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. Zbl 0956.11021. MR 1697859.

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Restricted product

See also

References