[Merged by Bors] - feat(topology/algebra/group): continuity of action of a group on its own coset space #11772
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…own coset space (#11772) Given a subgroup `Γ` of a topological group `G`, there is an induced scalar action of `G` on the coset space `G ⧸ Γ`, and there is also an induced topology on `G ⧸ Γ`. We prove that this action is continuous in each variable, and, if the group `G` is locally compact, also jointly continuous. Co-authored-by: Alex Kontorovich <58564076+AlexKontorovich@users.noreply.github.com>
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Given a subgroup
Γof a topological groupG, there is an induced scalar action ofGon the coset spaceG ⧸ Γ, and there is also an induced topology onG ⧸ Γ. We prove that this action is continuous in each variable, and, if the groupGis locally compact, also jointly continuous.Co-authored-by: Alex Kontorovich 58564076+AlexKontorovich@users.noreply.github.com
See Zulip for motivation of the oddly specific lemmas in
topology/compact_open. I am not sure whether the resultquotient_group.has_continuous_smultruly needs the hypothesis[locally_compact_space G]or whether this is an artifact of the proof method.