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

Let I {\displaystyle I} be an index set; S {\displaystyle S} a finite subset of I {\displaystyle I}. If G i {\displaystyle G_{i}} is a locally compact group for each i ∈ I {\displaystyle i\in I}, and K i ⊂ G i {\displaystyle K_{i}\subset G_{i}} is an open compact subgroup for each i ∈ I ∖ S {\displaystyle i\in I\setminus S}, then the restricted product

∏ i ′ G i {\displaystyle \prod _{i}\nolimits 'G_{i}\,}

is the subset of the product of the G i {\displaystyle G_{i}}'s consisting of all elements ( g i ) i ∈ I {\displaystyle (g_{i})_{i\in I}} such that g i ∈ K i {\displaystyle g_{i}\in K_{i}} for all but finitely many i ∈ I ∖ S {\displaystyle i\in I\setminus S}.

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

∏ i A i , {\displaystyle \prod _{i}A_{i}\,,}

where A i {\displaystyle A_{i}} is open in G i {\displaystyle G_{i}} and A i = K i {\displaystyle A_{i}=K_{i}} for all but finitely many i {\displaystyle 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.

See also