Restricted product

In mathematics, the restricted product is a construction in the theory of topological groups. Let [math]\displaystyle{ I }[/math] be an index set; [math]\displaystyle{ S }[/math] a finite subset of [math]\displaystyle{ I }[/math]. If [math]\displaystyle{ G_i }[/math] is a locally compact group for each [math]\displaystyle{ i \in I }[/math], and [math]\displaystyle{ K_i \subset G_i }[/math] is an open compact subgroup for each [math]\displaystyle{ i \in I \setminus S }[/math], then the restricted product

[math]\displaystyle{ \prod_i\nolimits' G_i\, }[/math]

is the subset of the product of the [math]\displaystyle{ G_i }[/math]'s consisting of all elements [math]\displaystyle{ (g_i)_{i \in I} }[/math] such that [math]\displaystyle{ g_i \in K_i }[/math] for all but finitely many [math]\displaystyle{ i \in I \setminus S }[/math].

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

[math]\displaystyle{ \prod_i A_i\,, }[/math]

where [math]\displaystyle{ A_i }[/math] is open in [math]\displaystyle{ G_i }[/math] and [math]\displaystyle{ A_i = K_i }[/math] for all but finitely many [math]\displaystyle{ i }[/math].

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

• Direct sum

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.