论文标题
密集分离的群体及其在$ d $独立中的应用
Dense-separable groups and its applications in $d$-independence
论文作者
论文摘要
如果拓扑空间称为{\ IT密集 - 独一}，则如果其每个致密子集都是可分离的。因此，每个密集的分离空间都是可分离的。我们建立了密集可分离的拓扑组的一些基本特性。我们证明，每个具有可计数紧密度的可分离空间都是茂密的，并且提供一个可分离的拓扑群，这在遗传性上不可分开。我们还证明，对于Hausdorff局部紧凑的群体而言，如果FF是可分离的，则是局部密集的。 此外，我们研究了密集的造成拓扑组。我们证明，对于每个紧凑型扭转（或可分裂或无扭转或完全断开的）abelian群体，如果是可分离的，则它是可分离的，如果是可分离的，则是可分离的。 最后，我们讨论了$ D $独立的拓扑组和相关结构中的某些应用。我们证明，每个常规的密集群 - 分离 - 可分离的Abelian半足组都有$ r_ {0}（g）\ geq \ mathfrak {c} $ is $ d $ notepentent。我们还证明，对于每个常规的密集群 - 分离的界有限的副学Abelian $ g $，带有$ | g |> 1 $，是$ d $ - 独立的Ifff iff，这是一个非$ m $ m $ -group iff iff iff if if每个非trivial trivial trivial priential optir $ g_ g_ {p} $ g $ g $ d $ d $ d $ d $ dipentent。应用此结果，我们证明可分开的可分离型副学Abelian $ g $，带有$ | g | = \ Mathfrak {C} $是$ D $ - 独立。此外，我们证明，具有非平凡连接组件的每个密集组可分离的地图Abelian组也是$ d $无关的。
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.
