w-Divisible groups

A topological abelian group G is w-divisible if G has uncountable weight and the subgroup m G = { m x : x ∈ G } has the same weight of G for each positive integer m. In order to “measure” w-divisibility we introduce a cardinal invariant (divisible weight) which allows for a precise description of various phenomena related to the subgroups of the compact abelian groups. We give several applications of these results.