Restricted product

In mathematics, the restricted product is a construction in the theory of topological groups.
Let formula_1 be an index set; formula_2 a finite subset of formula_1. If formula_4 is a locally compact group for each formula_5, and formula_6 is an open compact subgroup for each formula_7, then the restricted product 
is the subset of the product of the formula_9's consisting of all elements formula_10 such that formula_11 for all but finitely many formula_7. 
This group is given the topology whose basis of open sets are those of the form
where formula_14 is open in formula_4 and formula_16 for all but finitely many formula_17.
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.