论文标题
拓扑组的通用特性
Generic properties of topological groups
论文作者
论文摘要
我们研究了Baire类别意义上拓扑组的通用特性。首先，我们研究无数（离散）组。我们扩展了B. H. Neumann，H。Simmons和A. MacIntyre在代数封闭的组和单词问题上的经典结果。 I. Goldbring，S。E。Kunnawalkam和Y. Lodha证明了每个同构类别在无数（离散）组中都微不足道。相比之下，我们表明，无限（离散的）阿贝尔群体中有一个合并的同构类别。然后，我们转向紧凑的可迁移的亚伯利亚群体。我们使用Pontryagin二元性表明，紧凑型的Abelian群体中有一个相当的同构类别。我们讨论了它与无数（离散）案例的连接。最后，我们研究紧凑的可迁移群体。我们证明，通用的紧凑型Metrizable群体既没有连接，也没有完全断开连接。同样，它既不是无扭力的扭转组。
We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. I. Goldbring, S. E. Kunnawalkam and Y. Lodha proved that every isomorphism class is meager among countably infinite (discrete) groups. In contrast, we show that there is a comeager isomorphism class among countably infinite (discrete) abelian groups. Then we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite (discrete) case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.
