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# Dense-separable groups and its applications in $d$-independence

Nov 2022

A topological space is called {\it dense-separable} if each dense subset ofits is separable. Therefore, each dense-separable space is separable. Weestablish some basic properties of dense-separable topological groups. We provethat each separable space with a countable tightness is dense-separable, andgive a dense-separable topological group which is not hereditarily separable.We also prove that, for a Hausdorff locally compact group , it is locallydense-separable iff it is metrizable. Moreover, we study dense-subgroup-separable topological groups. We provethat, for each compact torsion (or divisible, or torsion-free, or totallydisconnected) abelian group, it is dense-subgroup-separable iff it isdense-separable iff it is metrizable. Finally, we discuss some applications in $d$-independent topological groupsand related structures. We prove that each regular dense-subgroup-separableabelian semitopological group with $r_{0}(G)\geq\mathfrak{c}$ is$d$-independent. We also prove that, for each regular dense-subgroup-separablebounded paratopological abelian group $G$ with $|G|>1$, it is $d$-independentiff 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 separablemetrizable almost torsion-free paratopological abelian group $G$ with$|G|=\mathfrak{c}$ is $d$-independent. Further, we prove that eachdense-subgroup-separable MAP abelian group with a nontrivial connectedcomponent is also $d$-independent.

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