Restricted product
In mathematics, the restricted product is a construction in the theory of topological groups.
Let <math>I</math> be an index set; <math>S</math> a finite subset of <math>I</math>. If <math>G_i</math> is a locally compact group for each <math>i \in I</math>, and <math>K_i \subset G_i</math> is an open compact subgroup for each <math>i \in I \setminus S</math>, then the restricted product
- <math>\prod_i\nolimits' G_i\,</math>
is the subset of the product of the <math> G_i </math>'s consisting of all elements <math>(g_i)_{i \in I}</math> such that <math>g_i \in K_i </math> for all but finitely many <math>i \in I \setminus S</math>.
This group is given the topology whose basis of open sets are those of the form
- <math>\prod_i A_i\,,</math>
where <math>A_i</math> is open in <math>G_i</math> and <math>A_i = K_i</math> for all but finitely many <math>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.