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.
restricted, product, mathematics, restricted, product, construction, theory, topological, groups, displaystyle, index, displaystyle, finite, subset, displaystyle, displaystyle, locally, compact, group, each, displaystyle, displaystyle, subset, open, compact, s. 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 editDirect sumReferences editFrohlich A Cassels J W 1967 Algebraic number theory Boston MA Academic Press ISBN 978 0 12 163251 9 Neukirch Jurgen 1999 Algebraische Zahlentheorie Grundlehren der mathematischen Wissenschaften Vol 322 Berlin Springer Verlag ISBN 978 3 540 65399 8 MR 1697859 Zbl 0956 11021 Retrieved from https en wikipedia org w index php title Restricted product amp oldid 1084813414, wikipedia, wiki, book, books, library,