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 r0β(G)β₯c is
d-independent. We also prove that, for each regular dense-subgroup-separable
bounded paratopological abelian group G with β£Gβ£>1, it is d-independent
iff it is a nontrivial M-group iff each nontrivial primary component Gpβ
of G is d-independent. Apply this result, we prove that a separable
metrizable almost torsion-free paratopological abelian group G with
β£Gβ£=c is d-independent. Further, we prove that each
dense-subgroup-separable MAP abelian group with a nontrivial connected
component is also d-independent.Comment: 19 page