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 s u ( X ) : = { x ∈ X : ( u n , x ) → 0 in T } . Every characterized subgroup is an F σ δ -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. roup 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. ery 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.