On a generalization of Abelian sequential groups

Saak S. Gabriyelyan¹

¹ Ben-Gurion University of the Negev P.O. Box 653 Be'er-Sheva 84105, Israel

Fundamenta Mathematicae, Tome 221 (2013) no. 2, pp. 95-127.

Résumé

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.

Détail

DOI : 10.4064/fm221-2-1

Keywords: tau hausdorff abelian topological group called s group resp bs group there set sequences tau finest hausdorff resp precompact group topology which every sequence converges zero characterizations abelian s bs groups given tau maximally almost periodic map abelian s group its pontryagin dual group tau wedge dense mathfrak closed subgroup compact group wedge where group discrete topology converse every dense mathfrak closed subgroup wedge there topology tau tau s group tau wedge algebraically proved locally compact non compact abelian group cardinality ulam measurable realcompact bs group s group where group endowed bohr topology every reflexive polish abelian group mathfrak closed its bohr compactification particular countable tau generated countable set convergent sequences shown dual group tau wedge polish abelian group called characterizable dual group countable abelian map s group whose topology generated sequence converging zero characterizable abelian group schwartz group locally compact dual group characterizable abelian group characterizable locally compact