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Abstract

Let $(G,\tau )$ be a Hausdorff 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 $\tau $ is the finest Hausdorff (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 $(G,\tau )$ is a maximally almost periodic (MAP) Abelian $s$-group, then its Pontryagin dual group $(G,\tau )^\wedge $ is a dense $\mathfrak {g}$-closed subgroup of the compact group $(G_d)^\wedge $, where $G_d$ is the group $G$ with the discrete topology. The converse is also true: for every dense $\mathfrak {g}$-closed subgroup $H$ of $(G_d)^\wedge $, there is a topology $\tau $ on $G$ such that $(G,\tau )$ is an $s$-group and $(G,\tau )^\wedge =H$ 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 reflexive Polish Abelian group is $\mathfrak {g}$-closed in its Bohr compactification. In the particular case when $G$ is countable and $\tau $ is generated by a countable set of convergent sequences, it is shown that the dual group $(G,\tau )^\wedge $ 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.