On T-characterized subgroups of compact Abelian groups

A sequence [u_n]n∈ωin abstract additively-written Abelian group G is called a T-sequence if there is a Hausdorff group topology on G relative to which lim_n u_n = 0. We say that a subgroup H of an infinite compact Abelian group X is T-characterized if there is a T-sequence u = [u_n] in the dual group of X, such that H = [x ∈ X: (u_n;, x) → 1]. We show that a closed subgroup H of X is T-characterized if and only if H is a GΔ-subgroup of X and the annihilator of H admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group X are T-characterized if and only if X is metrizable and connected. We prove that every compact Abelian group X of infinite exponent has a T-characterized subgroup, which is not an F_σ-subgroup of X, that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427-2442).