  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).
