On a generalization of Abelian sequential groups

Let (G; τ ) be a Hausdor Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdor (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G; τ ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group (G; τ )^λ is a dense g- closed subgroup of the compact group (Gd)^λ, where Gd is the group G with the discrete topology. The converse is also true: for every dense g-closed subgroup H of (Gd)^λ, there is a topology τ on G such that (G; τ ) is an s-group and (G; τ )^λ = H algebraically. It is proved that, if G is a locally compact non-compact Abelian group such that the cardinality |G| of G is not Ulam measurable, then G ⁺ is a realcompact bs-group that is not an s-group, where G ⁺ is the group G endowed with the Bohr topology. We show that every reexive Polish Abelian group is g-closed in its Bohr compactication. In the particular case when G is countable and τ is generated by a countable set of convergent sequences, it is shown that the dual group (G; τ ) ^λ is Polish. An Abelian group X is called characterizable if it is the dual group of a countable Abelian MAP s-group whose topology is generated by one sequence converging to zero. A characterizable Abelian group is a Schwartz group iff it is locally compact. The dual group of a characterizable Abelian group X is characterizable iff X is locally compact.