File: Mathlib/Topology/Algebra/Group/OpenMapping.lean

full_name: vadd_singleton_mem_nhds_of_sigmaCompact
code: /-- 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. -/
@[to_additive "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."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x 
goal: U +ᵥ {x} ∈ 𝓝 x

full_name: smul_singleton_mem_nhds_of_sigmaCompact
code: /-- 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. -/
@[to_additive "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."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x 
goal: U • {x} ∈ 𝓝 x

full_name: isOpenMap_vadd_of_sigmaCompact
code: /-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open. This is a version of the open mapping theorem, valid notably for the
action of a sigma-compact locally compact group on a locally compact space. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space."]
theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) 
goal: IsOpenMap fun g => g +ᵥ x

full_name: isOpenMap_smul_of_sigmaCompact
code: /-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open. This is a version of the open mapping theorem, valid notably for the
action of a sigma-compact locally compact group on a locally compact space. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space."]
theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) 
goal: IsOpenMap fun g => g • x

full_name: MonoidHom.isOpenMap_of_sigmaCompact
code: /-- A surjective morphism of topological groups is open when the source group is sigma-compact and
the target group is a Baire space (for instance a locally compact group). -/
@[to_additive]
theorem MonoidHom.isOpenMap_of_sigmaCompact
    {H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
    (f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
    IsOpenMap f 
goal: IsOpenMap ⇑f

full_name: AddMonoidHom.isOpenMap_of_sigmaCompact
code: /-- A surjective morphism of topological groups is open when the source group is sigma-compact and
the target group is a Baire space (for instance a locally compact group). -/
@[to_additive]
theorem MonoidHom.isOpenMap_of_sigmaCompact
    {H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
    (f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
    IsOpenMap f 
goal: IsOpenMap ⇑f