On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u=(u_n) of characters of X such that H=s_u(X), where su(X):={x∈X:(un,x)→0 in T}. Every characterized subgroup is an F_σδ-subgroup of X. We show that every G_δ-subgroup of X is characterized. On the other hand, X has non-characterized F_σ-subgroups.A subgroup H of X is said to be countable modulo compact (CMC) if H has a subgroup K such that it is a compact G_δ-subgroup of X and H/K is countable. It is proved that every characterized subgroup H of X is CMC if and only if X has finite exponent. This result gives a complete description of the characterized subgroups of compact abelian groups of finite exponent.For every sequence u=(u_n) of characters of X we define a refinement X_u of X, that is a Čech complete locally quasi-convex (almost metrizable) group. With the sequence u we associate the closed subgroup H_u of X_u and the natural projection π_X:X_u→X such that π_X(H_u)=s_u(X). This provides a description of the characterized subgroups of arbitrary compact abelian groups, extending the previously existing result from [25]. This description is new even in the case of metrizable compact groups.