On a generalization of Abelian sequential groups

Saak S. Gabriyelyan. "On a generalization of Abelian sequential groups." Fundamenta Mathematicae 221.2 (2013): 95-127. <http://eudml.org/doc/283287>.

@article{SaakS2013,
abstract = {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.},
author = {Saak S. Gabriyelyan},
journal = {Fundamenta Mathematicae},
keywords = {T -sequence; TB -sequence; abelian group; s-group; bs-group; compact abelian group; dual group; g-closed subgroup; sequentially covering map},
language = {eng},
number = {2},
pages = {95-127},
title = {On a generalization of Abelian sequential groups},
url = {http://eudml.org/doc/283287},
volume = {221},
year = {2013},
}

TY - JOUR
AU - Saak S. Gabriyelyan
TI - On a generalization of Abelian sequential groups
JO - Fundamenta Mathematicae
PY - 2013
VL - 221
IS - 2
SP - 95
EP - 127
AB - 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.
LA - eng
KW - T -sequence; TB -sequence; abelian group; s-group; bs-group; compact abelian group; dual group; g-closed subgroup; sequentially covering map
UR - http://eudml.org/doc/283287
ER -