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

TL;DR

The paper proves that every second countable abelian group can be embedded into a second countable divisible group, extending pseudonorms and topologies while relying on the Axiom of Choice.

Contribution

It introduces a method to extend pseudonorms and topologies from subgroups to entire groups, establishing embeddings into divisible groups under the Axiom of Choice.

Findings

Every separable metrizable abelian group is a subgroup of a divisible one.

Pseudonorms on subgroups can be extended to the whole group without changing density.

Results depend on the Axiom of Choice and do not hold under the Axiom of Determinacy.

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).