Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Abstract

Let G be compact abelian group such that w ( C ( G ) ) = w ( C ( G ) ) ω . We prove that if | C ( G ) | ⩾ m ( G / C ( G ) ) , then G contains a dense pseudocompact subgroup without non-trivial convergent sequences, where C ( G ) is the component of the identity of G and m ( G ) is the smallest cardinality of a dense pseudocompact subgroup of G. As a consequence we obtain the following: (1) Every compact connected abelian group of weight κ with κ = κ ω has a dense pseudocompact subgroup without non-trivial convergent sequences. (2) [GCH] Let G be a compact abelian group whose connected component has weight κ with κ = κ ω . The following assertions are then equivalent: (i) Every dense pseudocompact subgroup of G has a non-trivial convergent sequence. (ii) One of the following two conditions is satisfied: (a) For some n < ω , nG is infinite and cf ( w ( n G ) ) = ω . (b) | C ( G ) | < m ( log ( r 0 ( G / C ( G ) ) ) ) .