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

Restricted product

See also

References