Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties

Abstract

It is well-known ((2), Theorem 9·11) that any closed subgroup of Rn is isomorphic (topologically and algebraically) to Ra × Zb, where a, b are suitable non-negative integers. For an infinite product of copies of R, it is also known that any locally compact (hence closed) subgroup is a product of copies R and Z, and that any connected subgroup is a product of copies of R (see (7), (3), respectively). Some information is also given in (3) on closed subgroups of products of copies of R and T, where T = R/Z is the circle group.