On T -sequences and characterized subgroups

Let X be a compact metrizable abelian group and u = {un} be a sequence in its dual X. Set su(X) = {x : (un, x) → 1} and T H 0 = {(zn) ∈ T ∞ : zn → 1}. Let G be a subgroup of X. We prove that G = su(X) for some u iff it can be represented as some dually closed subgroup Gu of ClXG × T H 0 . In particular, su(X) is polishable. Let u = {un} be a T -sequence. Denote by (X̂,u) the group X ∧ equipped with the finest group topology in which un → 0. It is proved that (X̂,u) ∧ = Gu and n(X̂,u) = su(X) . We also prove that the group generated by a Kronecker set can not be characterized. We shall write our abelian groups additively. For a topological groupX, X̂ denotes the group of all continuous characters on X. We denote its dual group by X, i.e. the group X̂ endowed with the compact-open topology. A group X equipped with discrete topology is denoted by Xd. Denote by n(X) = ∩χ∈ b Xkerχ the von Neumann radical of X. X is named Pontryagin reflexive or reflexive if the canonical homomorphism αX : X → X , x 7→ (χ 7→ (χ, x)) is a topological isomorphism. If H is a subgroup of X, we denote by H its annihilator. Let X and Y be topological groups and φ : X → Y a continuous homomorphism. We denote by φ : Y ∧ → X, χ 7→ χ ◦ φ, the dual homomorphism of φ. Let A be a subset of a group X. Set (2)A = A+ A, (n+ 1)A = (n)A + A. 〈A〉 denotes the subgroup generated by A. Let G be a Borel subgroup of a Polish group X. G is called polishable if there exists a Polish group topology τ on G such that the identity map i : (G, τ) → X, i(g) = g, is continuous. We remaind that a Polish group topology on a Borel subgroup of a Polish group is unique. Let X be a compact metrizable group and u = {un} a sequence of elements of X̂. We denote by su(X) the set of all x ∈ X such that (un, x) → 1. Let G be a subgroup of X. If G = su(X) we say that u characterizes G and that G is characterized (by u) or basic g-closed. Denote by gX(G) := ⋂ {su(X) : u ∈ X̂ , G 6 su(X)}. G is called g-closed (resp. g-dense) if G = gX(G) (resp. gX(G) = X) [16]. The following group plays the key role in our considerations. Set [20] T H 0 := {ω = (zn) ∈ T ∞ : zn → 1} , d0(ω1, ω2) = sup{|z 1 n − z 2 n|, n ∈ N}. Then T0 is a Polish group and ( T H 0 )∧ = Z∞0 = {n = (n1, . . . , nk, 0, . . . ), nj ∈ Z}. The topology on Z∞0 is described in [20]. Note that T H 0 is reflexive and characterized subgroup of T ∞ by the sequence e1 = (1, 0, 0, . . . ), e2 = (0, 1, 0, . . . ), . . . The article is divided on two parts. ∗The author was partially supported by Israel Ministry of Immigrant Absorption