Each second countable abelian group is a subgroup of a second countable divisible group

zbMATH Open1102.22002arXiv0810.3030MaRDI QIDQ5487236

Lyubomyr Zdomskyy, Taras Banakh

Publication date: 19 September 2006

Abstract: It is shown that each pseudonorm defined on a subgroup $H$ of an abelian group $G$ can be extended to a pseudonorm on $G$ such that the densities of the obtained pseudometrizable topological groups coincide. We derive from this that any Hausdorff $omega$-bounded group topology on $H$ can be extended to a Hausdorff $omega$-bounded group topology on $G$. In its turn this result implies that each separable metrizable abelian group $H$ is a subgroup of a separable metrizable divisible group $G$. This result essentially relies on the Axiom of Choice and is not true under the Axiom of Determinacy (which contradicts to the Axiom of Choice but implies the Countable Axiom of Choice).

Full work available at URL: https://arxiv.org/abs/0810.3030

zbMATH Keywords

topological groupabelian groupdivisible groupaxiom of choiceaxiom of determinacypseudonormsecond countable abelian group

Mathematics Subject Classification ID

Structure of general topological groups (22A05) Topological groups (topological aspects) (54H11) Axiom of choice and related propositions (03E25) Topological methods for abelian groups (20K45) Determinacy principles (03E60)

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Universal countable-dimensional topological groups ⋮ Unnamed Item ⋮ Metrization criteria for compact groups in terms of their dense subgroups

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