发表机构
School of Mathematics and Computational Science, Wuyi University; School of Mathematical Sciences, Guangxi Minzu University（五邑大学数学与计算科学学院; 广西民族大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了两种Baire Pontryagin自反群，解决了预紧阿贝尔群可分商群的三个问题，否定了问题1.25，肯定回答了问题3.12，并实现了问题3.14的所有正则性。
AI 中文摘要
我们研究了Leiderman、Morris和Tkachenko在《以色列数学杂志》上发表的文献\ref{LMT}中提出的关于拓扑群可分商群的三个问题。首先，我们在ZFC中构造了$\bbT^{\bbC}$的一个连通的Baire Pontryagin自反稠密子群，其可数子群是$h$-嵌入的，而不可数子群是稠密的。其底层抽象群是圆周群，且其所有紧子集都是有限的。其次，我们构造了一个具有相同子群性质的零维Baire Pontryagin自反例子，其底层群是秩为$\bbC$的自由阿贝尔群。这两个例子都没有非平凡的可分Hausdorff商群。第三，对于他们在定理3.5中构造的群，我们确定了每个有限幂的每个闭子群（在坐标的整数变换意义下），并证明了每个有限幂的每个Hausdorff商群的每个可数子群都是$h$-嵌入且闭的。同样的结论也适用于我们的自由Baire自反例子。这些结果否定了问题1.25，肯定回答了问题3.12，并同时实现了文献\ref{LMT}中问题3.14的所有三个正则性性质。
英文摘要
We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.
