Restricted product

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

Let $$I$$ be an index set; $$S$$ a finite subset of $$I$$. If $$G_i$$ is a locally compact group for each $$i \in I$$, and $$K_i \subset G_i$$ is an open compact subgroup for each $$i \in I \setminus S$$, then the restricted product
 * $$\prod_i\nolimits' G_i\,$$

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 $$i \in I \setminus S$$.

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.