In mathematics, the restricted product is a construction in the theory of topological groups.
Let be an indexing set; a finite subset of . If for each, is a locally compact group, and for each, is an open compact subgroup, then the restricted product
is the subset of the product of the 's consisting of all elements such that for all but finitely many .
This group is given the topology whose basis of open sets are those of the form
where is open in and for all but finitely many .
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.
Famous quotes containing the words restricted and/or product:
“Language can only deal meaningfully with a special, restricted segment of reality. The rest, and it is presumably the much larger part, is silence.”
—George Steiner (b. 1929)
“Perhaps I am still very much of an American. That is to say, naïve, optimistic, gullible.... In the eyes of a European, what am I but an American to the core, an American who exposes his Americanism like a sore. Like it or not, I am a product of this land of plenty, a believer in superabundance, a believer in miracles.”
—Henry Miller (1891–1980)