On T-Characterized Subgroups of Compact Abelian Groups

A sequence {un}n∈ω in abstract additively-written Abelian group G is called a T -sequence if there is a Hausdorff group topology on G relative to which limn un = 0. We say that a subgroupH of an infinite compact Abelian groupX is T -characterized if there is a T -sequence u = {un} in the dual group of X , such that H = {x ∈ X : (un, x) → 1}. We show that a closed subgroupH ofX is T -characterized if and only ifH is aGδ-subgroup ofX and the annihilator ofH admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian groupX 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 Fσ-subgroup of X , that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442).