2022/11/26 by Lin, Fucai, Wu, Qiyun, Liu, Chuan
#20K25 #20K45 #54B05 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Primary 54D65 #Secondary 22A05
paper · doi:10.48550/arxiv.2211.14588
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 r0(G)≥\mathfrakc 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 Gp of G is d-independent. Apply this result, we prove that a separable metrizable almost torsion-free paratopological abelian group G with |G|=\mathfrakc is d-independent. Further, we prove that each dense-subgroup-separable MAP abelian group with a nontrivial connected component is also d-independent.