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 ∈ I , G i is a locally compact group, and for each i ∈ I ∖ S , K i ⊂ G i is an open compact subgroup, then the restricted product

is the subset of the product of the G i 's consisting of all elements ( g i ) i ∈ I such that g i ∈ K i for all but finitely many i ∈ I ∖ S .

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

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.