Theorems · Theorem · Lie groups

vadd_singleton_mem_nhds_of_sigmaCompact

∀ {G : Type u_1} {X : Type u_2} [inst : TopologicalSpace G] [inst_1 : TopologicalSpace X] [inst_2 : AddGroup G]
  [IsTopologicalAddGroup G] [inst_4 : AddAction G X] [SigmaCompactSpace G] [BaireSpace X] [T2Space X]
  [ContinuousVAdd G X] [AddAction.IsPretransitive G X] {U : Set G}, U ∈ nhds 0 → ∀ (x : X), U +ᵥ {x} ∈ nhds x

Consider a sigma-compact additive group acting continuously and transitively on a Baire space. Then the orbit map is open around zero. It follows in isOpenMap_vadd_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

TopologicalSpaceTopologicalSpaceAddGroupIsTopologicalAddGroupAddActionSigmaCompactSpaceBaireSpaceT2SpaceContinuousVAddAddAction.IsPretransitive