On T-sequences and characterized subgroups

Let X be a compact metrizable abelian group and u = { u n } be a sequence in its dual group X ∧ . Set s u ( X ) = { x : ( u n , x ) → 1 } and T 0 H = { ( z n ) ∈ T ∞ : z n → 1 } . Let G be a subgroup of X. We prove that G = s u ( X ) for some u iff it can be represented as some dually closed subgroup G u of Cl X G × T 0 H . In particular, s u ( X ) is polishable. Let u = { u n } be a T-sequence. Denote by ( X ˆ , u ) the group X ∧ equipped with the finest group topology in which u n → 0 . It is proved that ( X ˆ , u ) ∧ = G u and n ( X ˆ , u ) = s u ( X ) ⊥ . We also prove that the group generated by a Kronecker set cannot be characterized.