  A topological space is called {\it dense-separable} if each dense subset of
its is separable. Therefore, each dense-separable space is separable. We
establish some basic properties of dense-separable topological groups. We prove
that each separable space with a countable tightness is dense-separable, and
give a dense-separable topological group which is not hereditarily separable.
We also prove that, for a Hausdorff locally compact group , it is locally
dense-separable iff it is metrizable.
  Moreover, we study dense-subgroup-separable topological groups. We prove
that, for each compact torsion (or divisible, or torsion-free, or totally
disconnected) abelian group, it is dense-subgroup-separable iff it is
dense-separable iff it is metrizable.
  Finally, we discuss some applications in $d$-independent topological groups
and related structures. We prove that each regular dense-subgroup-separable
abelian semitopological group with $r_{0}(G)\geq\mathfrak{c}$ is
$d$-independent. We also prove that, for each regular dense-subgroup-separable
bounded paratopological abelian group $G$ with $|G|>1$, it is $d$-independent
iff it is a nontrivial $M$-group iff each nontrivial primary component $G_{p}$
of $G$ is $d$-independent. Apply this result, we prove that a separable
metrizable almost torsion-free paratopological abelian group $G$ with
$|G|=\mathfrak{c}$ is $d$-independent. Further, we prove that each
dense-subgroup-separable MAP abelian group with a nontrivial connected
component is also $d$-independent.
