AI 中文总结
该研究证明了多个拓扑空间的同胚群具有特定数量的极大子群，还给出了作用于拓扑空间的群拥有至少$2^{2^{\boldsymbol{\text{aleph}}_0}}$个极大子群的充分条件，并确定了赋予点态拓扑的两类同胚群的开、闭极大子群数量。
AI 中文摘要
我们证明以下空间的同胚群恰有$2^{2^{\boldsymbol{\text{aleph}}_0}}$个极大子群：有理数空间$\boldsymbol{\text{Q}}$、贝尔空间$\boldsymbol{\text{N}}^{\boldsymbol{\text{N}}}$、其中$2^{\boldsymbol{\text{N}}}$为康托集的空间$\boldsymbol{\text{N}} \times 2^{\boldsymbol{\text{N}}}$、带序拓扑的序数$\boldsymbol{\text{\textomega}}^2$以及索根弗雷直线$\boldsymbol{\text{S}}$。更一般地，我们找到了作用在拓扑空间上的群$\boldsymbol{G}$的充分条件，该条件意味着$\boldsymbol{G}$至少有$2^{2^{\boldsymbol{\text{aleph}}_0}}$个极大子群。此外，若给$\text{Homeo}(\boldsymbol{\text{Q}})$和$\text{Homeo}(\boldsymbol{\text{N}}^{\boldsymbol{\text{N}}})$赋予点态拓扑，则可证明$\text{Homeo}(\boldsymbol{\text{N}}^{\boldsymbol{\text{N}}})$恰有$2^{\boldsymbol{\text{aleph}}_0}$个开极大子群，而$\text{Homeo}(\boldsymbol{\text{Q}})$恰有$\boldsymbol{\text{aleph}}_0$个开极大子群和$2^{\boldsymbol{\text{aleph}}_0}$个闭极大子群。
英文摘要
We show that the homeomorphism groups of the following spaces have precisely $2^{2^{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}^{\mathbb{N}}$, the space $\mathbb{N}\times 2^{\mathbb{N}}$ where $2^{\mathbb{N}}$ is the Cantor set, the ordinal $ω^2$ under its order topology, and the Sorgenfrey line $\mathbb{S}$. More generally, we find sufficient conditions on a group $G$ acting on a topological space which imply that $G$ has at least $2^{2^{\aleph_0}}$ maximal subgroups. Moreover, if the groups $\operatorname{Homeo}(\mathbb{Q})$ and $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ are equipped with the pointwise topology, then it is shown that $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ has precisely $2^{\aleph_0}$ open maximal subgroups, and $\operatorname{Homeo}(\mathbb{Q})$ has precisely $\aleph_0$ open maximal subgroups and $2^{\aleph_0}$ closed maximal subgroups.
Comments15 pages
