Maximal subgroups of homeomorphism groups

Abstract: We show that the homeomorphism groups of the following spaces have precisely $2<sup>{2<sup>{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}<sup>{\mathbb{N}}$, the space $\mathbb{N}\times 2<sup>{\mathbb{N}}$ where $2<sup>{\mathbb{N}}$ is the Cantor set, the ordinal $ω<sup>2$ 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 $2<sup>{2<sup>{\aleph_0}}$ maximal subgroups. Moreover, if the groups $\operatorname{Homeo}(\mathbb{Q})$ and $\operatorname{Homeo}(\mathbb{N}<sup>\mathbb{N})$ are equipped with the pointwise topology, then it is shown that $\operatorname{Homeo}(\mathbb{N}<sup>\mathbb{N})$ has precisely $2<sup>{\aleph_0}$ open maximal subgroups, and $\operatorname{Homeo}(\mathbb{Q})$ has precisely $\aleph_0$ open maximal subgroups and $2<sup>{\aleph_0}$ closed maximal subgroups.