  A topological group $X$ is called $duoseparable$ if there exists a countable
set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the
unit. We construct a functor $F$ assigning to each (abelian) topological group
$X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an
isomorphic copy of $X$. In fact, the functor $F$ is defined on the category of
unital topologized magmas. Also we prove that each $\sigma$-compact locally
compact abelian topological group embeds into a duoseparable locally compact
abelian-by-countable topological group.
