On a generalization of Abelian sequential groups

Saak S. Gabriyelyan

Fundamenta Mathematicae, Tome 220 (2013), p. 95-127 / Harvested from The Polish Digital Mathematics Library

Résumé

Let (G,τ) 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 τ 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,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group $(G,τ)^{∧}$ is a dense -closed subgroup of the compact group $(G_d)^{∧}$, where $G_d$ is the group G with the discrete topology. The converse is also true: for every dense -closed subgroup H of $(G_d)^{∧}$, there is a topology τ on G such that (G,τ) is an s-group and $(G,τ)^{∧}= H$ 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 -closed in its Bohr compactification. 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 $(G,τ)^{∧}$ 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.

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Publié le : 2013-01-01

Zbl 06162110

EUDML-ID : urn:eudml:doc:283287