Theorems · Theorem · Lie groups

smul_singleton_mem_nhds_of_sigmaCompact

∀ {G : Type u_1} {X : Type u_2} [inst : TopologicalSpace G] [inst_1 : TopologicalSpace X] [inst_2 : Group G]
  [IsTopologicalGroup G] [inst_4 : MulAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X] [ContinuousSMul G X]
  [MulAction.IsPretransitive G X] {U : Set G}, U ∈ nhds 1 → ∀ (x : X), U • {x} ∈ nhds x

Consider a sigma-compact group acting continuously and transitively on a Baire space. Then the orbit map is open around the identity. It follows in isOpenMap_smul_of_sigmaCompact that it is open around any point.

Defined in

Mathlib.Topology.Algebra.Group.OpenMapping

Cited by

1 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceGroupIsTopologicalGroupMulActionSigmaCompactSpaceBaireSpaceT2SpaceContinuousSMulMulAction.IsPretransitive