On T-sequences and characterized subgroups

摘要

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_o~H = {(z_n) 6 T~∞: z_n →1). Let C be a subgroup of X. We prove that C = s_u(X) for some u iff it can be represented as some dually closed subgroup C_u of Clx G x T_O~H In particular, s_u(X) is polishable. Let u = {u_n) be a T-sequence. Denote by (X, u) the group X~^ equipped with the finest group topology in which u_n→ 0. It is proved that (X, u)~^ = C_u and n(X, u) = S_u(X)~⊥. We also prove that the group generated by a Kronecker set cannot be characterized.