ZFC中的华莱士问题与可数紧无挠阿贝尔群
The Wallace problem and countably compact torsion-free Abelian groups in ZFC
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中文总结 AI 辅助
该研究在ZFC公理体系下，证明基数为$\mathfrak c$的无挠阿贝尔群可赋予无非平凡收敛序列的豪斯多夫可数紧群拓扑，构造出满足双侧消去律但非群的可数紧拓扑半群，否定回答了华莱士问题，并衍生出相关拓扑结构结论。
中文摘要 AI 辅助
我们在ZFC公理体系中证明，每个基数为$\mathfrak c$的无挠阿贝尔群都可赋予一个无非平凡收敛序列的豪斯多夫可数紧群拓扑。这一结论尤其适用于自由阿贝尔群$\mathbb{Z}^{(\mathfrak c)}$、贝尔-斯佩克群$\mathbb{Z}^ω$以及$\mathbb{Q}^{(\mathfrak c)}$。对于在$\mathbb{Z}^{(\mathfrak c)}$上构造的拓扑，按坐标非负的锥在子空间拓扑下是可数紧的。由此可得，在ZFC中存在一个满足双侧消去律但并非群的交换吉洪诺夫可数紧拓扑半群，从而对华莱士问题给出了否定回答。结合已有结果，该主定理还在ZFC中得到一个包含双循环半群副本的吉洪诺夫可数紧拓扑半群，以及一个并非拓扑群的函数豪斯多夫可数紧仿拓扑群。
英文摘要
We prove in ZFC that every torsion-free Abelian group of cardinality $\mathfrak c$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences. In particular, this applies to the free Abelian group $\mathbb{Z}^{(\mathfrak c)}$, the Baer-Specker group $\mathbb{Z}^ω$ and $\mathbb{Q}^{(\mathfrak c)}$. For the topology constructed on $\mathbb{Z}^{(\mathfrak c)}$, the coordinatewise nonnegative cone is countably compact in the subspace topology. Consequently, there exists in ZFC a commutative Tychonoff countably compact topological semigroup which has two-sided cancellation but is not a group, giving a negative answer to Wallace's question. Combined with earlier results, the main theorem also yields in ZFC a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup and a functionally Hausdorff countably compact paratopological group that is not a topological group.
