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✨ Replace ContinuousMulAction with ContinuousConstSMul

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laurent-lost-commits
Shad Amethyst 11 months ago
parent a87dc079d6
commit 60111656b8
11 changed files with 147 additions and 264 deletions
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39
Rubin.lean
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@ -16,6 +16,7 @@ import Mathlib.Topology.Bases
import Mathlib.Topology.Compactness.Compact import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.Separation import Mathlib.Topology.Separation
import Mathlib.Topology.Homeomorph import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Algebra.ConstMulAction
import Rubin.Tactic import Rubin.Tactic
import Rubin.MulActionExt import Rubin.MulActionExt
@ -70,7 +71,7 @@ variable {G α : Type _}
variable [TopologicalSpace α] variable [TopologicalSpace α]
variable [Group G] variable [Group G]
variable [MulAction G α] variable [MulAction G α]
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
variable [FaithfulSMul G α] variable [FaithfulSMul G α]
-- TODO: modify the proof to be less "let everything"-y, especially the first half -- TODO: modify the proof to be less "let everything"-y, especially the first half
@ -83,7 +84,7 @@ lemma proposition_1_1_1 [h_lm : LocallyMoving G α] [T2Space α] (f g : G) (supp
have hx_ne_x := mem_support.mp x_in_supp_h have hx_ne_x := mem_support.mp x_in_supp_h
-- Since α is Hausdoff, there is a nonempty V ⊆ Support α f, with h •'' V disjoint from V -- Since α is Hausdoff, there is a nonempty V ⊆ Support α f, with h •'' V disjoint from V
have ⟨V, V_open, x_in_V, V_in_support, disjoint_img_V⟩ := disjoint_nbhd_in (support_open f) x_in_supp_f hx_ne_x have ⟨V, V_open, x_in_V, V_in_support, disjoint_img_V⟩ := disjoint_nbhd_in (support_isOpen f) x_in_supp_f hx_ne_x
-- let f₂ be a nontrivial element of the RigidStabilizer G V -- let f₂ be a nontrivial element of the RigidStabilizer G V
let ⟨f₂, f₂_in_rist_V, f₂_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem x_in_V) let ⟨f₂, f₂_in_rist_V, f₂_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem x_in_V)
@ -204,7 +205,7 @@ by
have five_points : Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := by have five_points : Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := by
apply moves_1234_of_moves_12 apply moves_1234_of_moves_12
exact (Set.inter_subset_right _ _ x_in_U) exact (Set.inter_subset_right _ _ x_in_U)
have U_open: IsOpen U := (IsOpen.inter (support_open f) (support_open (g^12))) have U_open: IsOpen U := (IsOpen.inter (support_isOpen f) (support_isOpen (g^12)))
let ⟨V₀, V₀_open, x_in_V₀, V₀_in_support, disjoint_img_V₀⟩ := disjoint_nbhd_in U_open x_in_U fx_moves let ⟨V₀, V₀_open, x_in_V₀, V₀_in_support, disjoint_img_V₀⟩ := disjoint_nbhd_in U_open x_in_U fx_moves
let ⟨V₁, V₁_open, x_in_V₁, disjoint_img_V₁⟩ := disjoint_nbhd_fin five_points let ⟨V₁, V₁_open, x_in_V₁, disjoint_img_V₁⟩ := disjoint_nbhd_fin five_points
@ -350,7 +351,7 @@ end AlgebraicDisjointness
section RegularSupport section RegularSupport
lemma lemma_2_2 (G: Type _) {α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] lemma lemma_2_2 (G: Type _) {α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
[ContinuousMulAction G α] [FaithfulSMul G α] [ContinuousConstSMul G α] [FaithfulSMul G α]
[T2Space α] [h_lm : LocallyMoving G α] [T2Space α] [h_lm : LocallyMoving G α]
{U : Set α} (U_open : IsOpen U) (U_nonempty : Set.Nonempty U) : {U : Set α} (U_open : IsOpen U) (U_nonempty : Set.Nonempty U) :
Monoid.exponent G•[U] = 0 := Monoid.exponent G•[U] = 0 :=
@ -487,7 +488,7 @@ by
-- we construct an open subset within `Support α h \ RegularSupport α f`, -- we construct an open subset within `Support α h \ RegularSupport α f`,
-- and we show that it is non-empty, open and (by construction) disjoint from `Support α f`. -- and we show that it is non-empty, open and (by construction) disjoint from `Support α f`.
lemma open_set_from_supp_not_subset_rsupp {G α : Type _} lemma open_set_from_supp_not_subset_rsupp {G α : Type _}
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [T2Space α] [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] [T2Space α]
{f h : G} (not_support_subset_rsupp : ¬Support α h ⊆ RegularSupport α f): {f h : G} (not_support_subset_rsupp : ¬Support α h ⊆ RegularSupport α f):
∃ V : Set α, V ⊆ Support α h ∧ Set.Nonempty V ∧ IsOpen V ∧ Disjoint V (Support α f) := ∃ V : Set α, V ⊆ Support α h ∧ Set.Nonempty V ∧ IsOpen V ∧ Disjoint V (Support α f) :=
by by
@ -497,7 +498,7 @@ by
rw [Set.diff_eq_compl_inter] rw [Set.diff_eq_compl_inter]
apply IsOpen.inter apply IsOpen.inter
· simp · simp
· exact support_open _ · exact support_isOpen _
have U_subset_supp_h : U ⊆ Support α h := by simp; apply Set.diff_subset have U_subset_supp_h : U ⊆ Support α h := by simp; apply Set.diff_subset
have U_disj_supp_f : Disjoint U (Support α f) := by have U_disj_supp_f : Disjoint U (Support α f) := by
apply Set.disjoint_of_subset_right apply Set.disjoint_of_subset_right
@ -526,7 +527,7 @@ by
apply subset_from_diff_closure_eq_empty apply subset_from_diff_closure_eq_empty
· apply regularSupport_regular · apply regularSupport_regular
· exact support_open _ · exact support_isOpen _
· rw [Set.not_nonempty_iff_eq_empty] at U_empty · rw [Set.not_nonempty_iff_eq_empty] at U_empty
exact U_empty exact U_empty
@ -543,7 +544,7 @@ by
lemma proposition_2_1 {G α : Type _} lemma proposition_2_1 {G α : Type _}
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [T2Space α] [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] [T2Space α]
[LocallyMoving G α] [h_faithful : FaithfulSMul G α] [LocallyMoving G α] [h_faithful : FaithfulSMul G α]
(f : G) : (f : G) :
AlgebraicCentralizer f = G•[RegularSupport α f] := AlgebraicCentralizer f = G•[RegularSupport α f] :=
@ -575,7 +576,7 @@ by
exact interior_subset exact interior_subset
· rfl · rfl
· rw [<-g_eq_g'] · rw [<-g_eq_g']
exact Disjoint.closure_left supp_disj (support_open _) exact Disjoint.closure_left supp_disj (support_isOpen _)
} }
intro h_in_centralizer intro h_in_centralizer
@ -630,7 +631,7 @@ by
-- Small lemma for remark 2.3 -- Small lemma for remark 2.3
theorem rigidStabilizer_inter_bot_iff_regularSupport_disj {G α : Type _} theorem rigidStabilizer_inter_bot_iff_regularSupport_disj {G α : Type _}
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [LocallyMoving G α] [FaithfulSMul G α] [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] [LocallyMoving G α] [FaithfulSMul G α]
{f g : G} : {f g : G} :
G•[RegularSupport α f] ⊓ G•[RegularSupport α g] = ⊥ G•[RegularSupport α f] ⊓ G•[RegularSupport α g] = ⊥
↔ Disjoint (RegularSupport α f) (RegularSupport α g) := ↔ Disjoint (RegularSupport α f) (RegularSupport α g) :=
@ -670,7 +671,7 @@ by
variable {G α : Type _} variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] [T2Space α] variable [TopologicalSpace α] [T2Space α]
variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] variable [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
/-- /--
This demonstrates that the disjointness of the supports of two elements `f` and `g` This demonstrates that the disjointness of the supports of two elements `f` and `g`
@ -685,7 +686,7 @@ lemma remark_2_3 {f g : G} :
(AlgebraicCentralizer f) ⊓ (AlgebraicCentralizer g) = ⊥ → Disjoint (Support α f) (Support α g) := (AlgebraicCentralizer f) ⊓ (AlgebraicCentralizer g) = ⊥ → Disjoint (Support α f) (Support α g) :=
by by
intro alg_disj intro alg_disj
rw [disjoint_interiorClosure_iff (support_open _) (support_open _)] rw [disjoint_interiorClosure_iff (support_isOpen _) (support_isOpen _)]
simp simp
repeat rw [<-RegularSupport.def] repeat rw [<-RegularSupport.def]
rw [<-rigidStabilizer_inter_bot_iff_regularSupport_disj] rw [<-rigidStabilizer_inter_bot_iff_regularSupport_disj]
@ -738,7 +739,7 @@ section HomeoGroup
open Topology open Topology
variable {G α : Type _} [Group G] [TopologicalSpace α] [T2Space α] variable {G α : Type _} [Group G] [TopologicalSpace α] [T2Space α]
variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] variable [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
example : TopologicalSpace G := TopologicalSpace.generateFrom (RigidStabilizerBasis.asSets G α) example : TopologicalSpace G := TopologicalSpace.generateFrom (RigidStabilizerBasis.asSets G α)
@ -772,7 +773,7 @@ section Ultrafilter
variable {G α : Type _} variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] [T2Space α] variable [TopologicalSpace α] [T2Space α]
variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] variable [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
def RSuppSubsets {α : Type _} [TopologicalSpace α] (V : Set α) : Set (Set α) := def RSuppSubsets {α : Type _} [TopologicalSpace α] (V : Set α) : Set (Set α) :=
{W ∈ RegularSupportBasis.asSet α | W ⊆ V} {W ∈ RegularSupportBasis.asSet α | W ⊆ V}
@ -801,7 +802,7 @@ by
lemma compact_subset_of_rsupp_basis (G : Type _) {α : Type _} lemma compact_subset_of_rsupp_basis (G : Type _) {α : Type _}
[Group G] [TopologicalSpace α] [T2Space α] [Group G] [TopologicalSpace α] [T2Space α]
[MulAction G α] [ContinuousMulAction G α] [MulAction G α] [ContinuousConstSMul G α]
[LocallyCompactSpace α] [HasNoIsolatedPoints α] [LocallyDense G α] [LocallyCompactSpace α] [HasNoIsolatedPoints α] [LocallyDense G α]
(U : RegularSupportBasis α): (U : RegularSupportBasis α):
∃ V : RegularSupportBasis α, (closure V.val) ⊆ U.val ∧ IsCompact (closure V.val) := ∃ V : RegularSupportBasis α, (closure V.val) ⊆ U.val ∧ IsCompact (closure V.val) :=
@ -1018,7 +1019,7 @@ end Ultrafilter
variable {G α : Type _} variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] variable [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α]
def IsRigidSubgroup (S : Set G) := def IsRigidSubgroup (S : Set G) :=
S ≠ {1} ∧ ∃ T : Finset G, S = ⨅ (f ∈ T), AlgebraicCentralizer f S ≠ {1} ∧ ∃ T : Finset G, S = ⨅ (f ∈ T), AlgebraicCentralizer f
@ -1085,7 +1086,7 @@ theorem IsRigidSubgroup.algebraicCentralizerBasis_val {S : Set G} (S_rigid : IsR
section toRegularSupportBasis section toRegularSupportBasis
variable (α : Type _) variable (α : Type _)
variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] variable [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α]
variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α] variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α]
theorem IsRigidSubgroup.has_regularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : theorem IsRigidSubgroup.has_regularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) :
@ -1204,7 +1205,7 @@ section Equivalence
open Topology open Topology
variable (α : Type _) variable (α : Type _)
variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] variable [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α]
variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α] variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α]
-- TODO: either see whether we actually need this step, or change these names to something memorable -- TODO: either see whether we actually need this step, or change these names to something memorable
@ -1383,7 +1384,7 @@ def RubinSpace (G : Type _) [Group G] := Quotient (RubinFilterSetoid G)
variable {β : Type _} variable {β : Type _}
variable [TopologicalSpace β] [MulAction G β] [ContinuousMulAction G β] variable [TopologicalSpace β] [MulAction G β] [ContinuousConstSMul G β]
instance : TopologicalSpace (RubinSpace G) := sorry instance : TopologicalSpace (RubinSpace G) := sorry

3
Rubin/AlgebraicDisjointness.lean
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@ -3,6 +3,7 @@ import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.Subgroup.Actions import Mathlib.GroupTheory.Subgroup.Actions
import Mathlib.GroupTheory.Commutator import Mathlib.GroupTheory.Commutator
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Algebra.ConstMulAction
import Mathlib.Data.Fintype.Perm import Mathlib.Data.Fintype.Perm
import Mathlib.Tactic.FinCases import Mathlib.Tactic.FinCases
import Mathlib.Tactic.IntervalCases import Mathlib.Tactic.IntervalCases
@ -123,7 +124,7 @@ variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] variable [TopologicalSpace α]
variable [MulAction G α] variable [MulAction G α]
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
variable [FaithfulSMul G α] variable [FaithfulSMul G α]
-- Kind of a boring lemma but okay -- Kind of a boring lemma but okay

145
Rubin/HomeoGroup.lean
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@ -1,6 +1,7 @@
import Mathlib.Logic.Equiv.Defs import Mathlib.Logic.Equiv.Defs
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Homeomorph import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Algebra.ConstMulAction
import Rubin.LocallyDense import Rubin.LocallyDense
import Rubin.Topology import Rubin.Topology
@ -119,8 +120,8 @@ instance homeoGroup_mulAction₁ : MulAction (HomeoGroup α) α where
intro f g x intro f g x
rfl rfl
instance homeoGroup_mulAction₁_continuous : Rubin.ContinuousMulAction (HomeoGroup α) α where instance homeoGroup_mulAction₁_continuous : ContinuousConstSMul (HomeoGroup α) α where
continuous := by continuous_const_smul := by
intro h intro h
constructor constructor
intro S S_open intro S S_open
@ -143,24 +144,21 @@ instance homeoGroup_mulAction₁_faithful : FaithfulSMul (HomeoGroup α) α wher
theorem HomeoGroup.smulImage_eq_image (g : HomeoGroup α) (S : Set α) : theorem HomeoGroup.smulImage_eq_image (g : HomeoGroup α) (S : Set α) :
g •'' S = ⇑g.toHomeomorph '' S := rfl g •'' S = ⇑g.toHomeomorph '' S := rfl
section ContinuousMulActionCoe section FromContinuousConstSMul
variable {G : Type _} [Group G] variable {G : Type _} [Group G]
variable [MulAction G α] [Rubin.ContinuousMulAction G α] variable [MulAction G α] [ContinuousConstSMul G α]
/-- /--
`fromContinuous` is a structure-preserving transformation from a continuous `MulAction` to a `HomeoGroup` `fromContinuous` is a structure-preserving transformation from a continuous `MulAction` to a `HomeoGroup`
--/ --/
def HomeoGroup.fromContinuous (α : Type _) [TopologicalSpace α] [MulAction G α] [Rubin.ContinuousMulAction G α] def HomeoGroup.fromContinuous (α : Type _) [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α]
(g : G) : HomeoGroup α := (g : G) : HomeoGroup α :=
HomeoGroup.from (Rubin.ContinuousMulAction.toHomeomorph α g) HomeoGroup.from (Homeomorph.smul g)
@[simp] @[simp]
theorem HomeoGroup.fromContinuous_def (g : G) : theorem HomeoGroup.fromContinuous_def (g : G) :
HomeoGroup.from (Rubin.ContinuousMulAction.toHomeomorph α g) = HomeoGroup.fromContinuous α g := rfl HomeoGroup.from (Homeomorph.smul g) = HomeoGroup.fromContinuous α g := rfl
-- instance homeoGroup_coe_fromContinuous : Coe G (HomeoGroup α) where
-- coe := fun g => HomeoGroup.fromContinuous α g
@[simp] @[simp]
theorem HomeoGroup.fromContinuous_smul (g : G) : theorem HomeoGroup.fromContinuous_smul (g : G) :
@ -169,7 +167,7 @@ by
intro x intro x
unfold fromContinuous unfold fromContinuous
rw [<-HomeoGroup.smul₁_def', HomeoGroup.from_toHomeomorph] rw [<-HomeoGroup.smul₁_def', HomeoGroup.from_toHomeomorph]
unfold Rubin.ContinuousMulAction.toHomeomorph unfold Homeomorph.smul
simp simp
theorem HomeoGroup.fromContinuous_one : theorem HomeoGroup.fromContinuous_one :
@ -216,7 +214,8 @@ by
repeat rw [Rubin.mem_support] repeat rw [Rubin.mem_support]
rw [<-HomeoGroup.smul₁_def, <-HomeoGroup.fromContinuous_def] rw [<-HomeoGroup.smul₁_def, <-HomeoGroup.fromContinuous_def]
rw [HomeoGroup.from_toHomeomorph] rw [HomeoGroup.from_toHomeomorph]
rw [Rubin.ContinuousMulAction.toHomeomorph_toFun] rw [Homeomorph.smul]
simp only [Equiv.toFun_as_coe, MulAction.toPerm_apply]
@[simp] @[simp]
theorem HomeoGroup.fromContinuous_regularSupport (g : G) : theorem HomeoGroup.fromContinuous_regularSupport (g : G) :
@ -232,7 +231,7 @@ by
repeat rw [Rubin.smulImage_def] repeat rw [Rubin.smulImage_def]
simp simp
def HomeoGroup.fromContinuous_embedding (α : Type _) [TopologicalSpace α] [MulAction G α] [Rubin.ContinuousMulAction G α] [FaithfulSMul G α]: G ↪ (HomeoGroup α) where def HomeoGroup.fromContinuous_embedding (α : Type _) [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α]: G ↪ (HomeoGroup α) where
toFun := fun (g : G) => HomeoGroup.fromContinuous α g toFun := fun (g : G) => HomeoGroup.fromContinuous α g
inj' := by inj' := by
intro g h fromCont_eq intro g h fromCont_eq
@ -245,7 +244,7 @@ def HomeoGroup.fromContinuous_embedding (α : Type _) [TopologicalSpace α] [Mul
theorem HomeoGroup.fromContinuous_embedding_toFun [FaithfulSMul G α] (g : G): theorem HomeoGroup.fromContinuous_embedding_toFun [FaithfulSMul G α] (g : G):
HomeoGroup.fromContinuous_embedding α g = HomeoGroup.fromContinuous α g := rfl HomeoGroup.fromContinuous_embedding α g = HomeoGroup.fromContinuous α g := rfl
def HomeoGroup.fromContinuous_monoidHom (α : Type _) [TopologicalSpace α] [MulAction G α] [Rubin.ContinuousMulAction G α] [FaithfulSMul G α]: G →* (HomeoGroup α) where def HomeoGroup.fromContinuous_monoidHom (α : Type _) [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α]: G →* (HomeoGroup α) where
toFun := fun (g : G) => HomeoGroup.fromContinuous α g toFun := fun (g : G) => HomeoGroup.fromContinuous α g
map_one' := by map_one' := by
simp simp
@ -255,120 +254,4 @@ def HomeoGroup.fromContinuous_monoidHom (α : Type _) [TopologicalSpace α] [Mul
intros intros
rw [fromContinuous_mul] rw [fromContinuous_mul]
-- theorem HomeoGroup.fromContinuous_rigidStabilizer (U : Set α) [FaithfulSMul G α]: end FromContinuousConstSMul
-- Rubin.RigidStabilizer (HomeoGroup α) U = Subgroup.map (HomeoGroup.fromContinuous_monoidHom α) (Rubin.RigidStabilizer G U) :=
-- by
-- ext g
-- rw [<-Subgroup.mem_carrier]
-- unfold Rubin.RigidStabilizer
-- simp
-- sorry
end ContinuousMulActionCoe
namespace Rubin
section Other
-- TODO: move this somewhere else
/--
## Proposition 3.1
--/
theorem rigidStabilizer_subset_iff (G : Type _) {α : Type _} [Group G] [TopologicalSpace α]
[MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α]
[h_lm : LocallyMoving G α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
U ⊆ V ↔ RigidStabilizer G U ≤ RigidStabilizer G V :=
by
constructor
exact rigidStabilizer_mono
intro rist_ss
by_contra U_not_ss_V
let W := U \ closure V
have W_nonempty : Set.Nonempty W := by
by_contra W_empty
apply U_not_ss_V
apply subset_from_diff_closure_eq_empty
· assumption
· exact U_reg.isOpen
· rw [Set.not_nonempty_iff_eq_empty] at W_empty
exact W_empty
have W_ss_U : W ⊆ U := by
simp
exact Set.diff_subset _ _
have W_open : IsOpen W := by
unfold_let
rw [Set.diff_eq_compl_inter]
apply IsOpen.inter
simp
exact U_reg.isOpen
have ⟨f, f_in_ristW, f_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open W_nonempty
have f_in_ristU : f ∈ RigidStabilizer G U := by
exact rigidStabilizer_mono W_ss_U f_in_ristW
have f_notin_ristV : f ∉ RigidStabilizer G V := by
apply rigidStabilizer_compl f_ne_one
apply rigidStabilizer_mono _ f_in_ristW
calc
W = U ∩ (closure V)ᶜ := by unfold_let; rw [Set.diff_eq_compl_inter, Set.inter_comm]
_ ⊆ (closure V)ᶜ := Set.inter_subset_right _ _
_ ⊆ Vᶜ := by
rw [Set.compl_subset_compl]
exact subset_closure
exact f_notin_ristV (rist_ss f_in_ristU)
theorem rigidStabilizer_eq_iff (G : Type _) [Group G] {α : Type _} [TopologicalSpace α]
[MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
RigidStabilizer G U = RigidStabilizer G V ↔ U = V :=
by
constructor
· intro rist_eq
apply le_antisymm <;> simp only [Set.le_eq_subset]
all_goals {
rw [rigidStabilizer_subset_iff G] <;> try assumption
rewrite [rist_eq]
rfl
}
· intro H_eq
rw [H_eq]
theorem homeoGroup_rigidStabilizer_subset_iff {α : Type _} [TopologicalSpace α]
[h_lm : LocallyMoving (HomeoGroup α) α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
U ⊆ V ↔ RigidStabilizer (HomeoGroup α) U ≤ RigidStabilizer (HomeoGroup α) V :=
by
rw [rigidStabilizer_subset_iff (HomeoGroup α) U_reg V_reg]
theorem homeoGroup_rigidStabilizer_eq_iff {α : Type _} [TopologicalSpace α]
[LocallyMoving (HomeoGroup α) α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
RigidStabilizer (HomeoGroup α) U = RigidStabilizer (HomeoGroup α) V ↔ U = V :=
by
constructor
· intro rist_eq
apply le_antisymm <;> simp only [Set.le_eq_subset]
all_goals {
rw [homeoGroup_rigidStabilizer_subset_iff] <;> try assumption
rewrite [rist_eq]
rfl
}
· intro H_eq
rw [H_eq]
theorem homeoGroup_rigidStabilizer_injective {α : Type _} [TopologicalSpace α] [LocallyMoving (HomeoGroup α) α]
: Function.Injective (fun U : { S : Set α // Regular S } => RigidStabilizer (HomeoGroup α) U.val) :=
by
intro ⟨U, U_reg⟩
intro ⟨V, V_reg⟩
simp
exact (homeoGroup_rigidStabilizer_eq_iff U_reg V_reg).mp
end Other
end Rubin

16
Rubin/InteriorClosure.lean
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@ -226,21 +226,15 @@ Set.Finite.induction_on' S_finite (by simp) (by
· exact IH · exact IH
) )
theorem interiorClosure_smulImage' {G : Type _} [Group G] [MulAction G α] theorem interiorClosure_smulImage {G : Type _} [Group G] [MulAction G α] [ContinuousConstSMul G α]
(g : G) (U : Set α) (g : G) (U : Set α) :
(g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)) :
InteriorClosure (g •'' U) = g •'' InteriorClosure U := InteriorClosure (g •'' U) = g •'' InteriorClosure U :=
by by
simp simp
rw [<-smulImage_interior' _ _ g_continuous] rw [<-smulImage_interior]
rw [<-smulImage_closure' _ _ g_continuous] rw [<-smulImage_closure]
theorem interiorClosure_smulImage {G : Type _} [Group G] [MulAction G α] [ContinuousMulAction G α]
(g : G) (U : Set α) :
InteriorClosure (g •'' U) = g •'' InteriorClosure U :=
interiorClosure_smulImage' g U (continuousMulAction_elem_continuous α g)
theorem smulImage_regular {G : Type _} [Group G] [MulAction G α] [ContinuousMulAction G α] theorem smulImage_regular {G : Type _} [Group G] [MulAction G α] [ContinuousConstSMul G α]
(g : G) (U : Set α) : (g : G) (U : Set α) :
Regular U ↔ Regular (g •'' U) := Regular U ↔ Regular (g •'' U) :=
by by

84
Rubin/LocallyDense.lean
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@ -1,8 +1,10 @@
import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Algebra.ConstMulAction
import Rubin.RigidStabilizer import Rubin.RigidStabilizer
import Rubin.InteriorClosure
namespace Rubin namespace Rubin
@ -121,7 +123,7 @@ variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] variable [TopologicalSpace α]
variable [MulAction G α] variable [MulAction G α]
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
variable [FaithfulSMul G α] variable [FaithfulSMul G α]
instance dense_locally_moving [T2Space α] instance dense_locally_moving [T2Space α]
@ -143,10 +145,8 @@ by
let U := (g⁻¹ •'' V) ∩ W let U := (g⁻¹ •'' V) ∩ W
use U use U
constructor constructor
{ exact IsOpen.inter (smulImage_isOpen g⁻¹ V_open) W_open
-- NOTE: if this is common, then we should make a tactic for solving IsOpen goals
exact IsOpen.inter (img_open_open g⁻¹ V V_open) W_open
}
constructor constructor
{ {
simp simp
@ -192,5 +192,79 @@ by
exact Set.inter_subset_left _ _ exact Set.inter_subset_left _ _
· exact disjoint_W_img · exact disjoint_W_img
/--
## Proposition 3.1:
If a group action is faithful, continuous and "locally moving",
then `U ⊆ V` if and only if `G•[U] ≤ G•[V]` when `U` and `V` are regular.
--/
theorem rigidStabilizer_subset_iff (G : Type _) {α : Type _} [Group G] [TopologicalSpace α]
[MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α]
[h_lm : LocallyMoving G α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
U ⊆ V ↔ G•[U] ≤ G•[V] :=
by
constructor
exact rigidStabilizer_mono
intro rist_ss
by_contra U_not_ss_V
let W := U \ closure V
have W_nonempty : Set.Nonempty W := by
by_contra W_empty
apply U_not_ss_V
apply subset_from_diff_closure_eq_empty
· assumption
· exact U_reg.isOpen
· rw [Set.not_nonempty_iff_eq_empty] at W_empty
exact W_empty
have W_ss_U : W ⊆ U := by
simp
exact Set.diff_subset _ _
have W_open : IsOpen W := by
unfold_let
rw [Set.diff_eq_compl_inter]
apply IsOpen.inter
simp
exact U_reg.isOpen
have ⟨f, f_in_ristW, f_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open W_nonempty
have f_in_ristU : f ∈ RigidStabilizer G U := by
exact rigidStabilizer_mono W_ss_U f_in_ristW
have f_notin_ristV : f ∉ RigidStabilizer G V := by
apply rigidStabilizer_compl f_ne_one
apply rigidStabilizer_mono _ f_in_ristW
calc
W = U ∩ (closure V)ᶜ := by unfold_let; rw [Set.diff_eq_compl_inter, Set.inter_comm]
_ ⊆ (closure V)ᶜ := Set.inter_subset_right _ _
_ ⊆ Vᶜ := by
rw [Set.compl_subset_compl]
exact subset_closure
exact f_notin_ristV (rist_ss f_in_ristU)
/--
A corollary of the previous theorem is that the rigid stabilizers of two regular sets `U` and `V`
are equal if and only if `U = V`.
--/
theorem rigidStabilizer_eq_iff (G : Type _) [Group G] {α : Type _} [TopologicalSpace α]
[MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
G•[U] = G•[V] ↔ U = V :=
by
constructor
· intro rist_eq
apply le_antisymm <;> simp only [Set.le_eq_subset]
all_goals {
rw [rigidStabilizer_subset_iff G] <;> try assumption
rewrite [rist_eq]
rfl
}
· intro H_eq
rw [H_eq]
end Rubin end Rubin

4
Rubin/RegularSupport.lean
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@ -48,13 +48,13 @@ variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] variable [TopologicalSpace α]
variable [MulAction G α] variable [MulAction G α]
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
theorem support_subset_regularSupport [T2Space α] {g : G} : theorem support_subset_regularSupport [T2Space α] {g : G} :
Support α g ⊆ RegularSupport α g := Support α g ⊆ RegularSupport α g :=
by by
apply interiorClosure_subset apply interiorClosure_subset
apply support_open (α := α) (g := g) apply support_isOpen (α := α) (g := g)
#align support_in_regular_support Rubin.support_subset_regularSupport #align support_in_regular_support Rubin.support_subset_regularSupport
theorem regularSupport_subset {g : G} {U : Set α} : theorem regularSupport_subset {g : G} {U : Set α} :

3
Rubin/RegularSupportBasis.lean
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@ -5,6 +5,7 @@ made up of finite intersections of `RegularSupport α g` for `g : HomeoGroup α`
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Homeomorph import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Algebra.ConstMulAction
import Rubin.LocallyDense import Rubin.LocallyDense
import Rubin.Topology import Rubin.Topology
@ -349,7 +350,7 @@ open Topology
variable (G α : Type _) variable (G α : Type _)
variable [Group G] variable [Group G]
variable [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] [HasNoIsolatedPoints α] variable [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] [HasNoIsolatedPoints α]
variable [MulAction G α] [LocallyDense G α] [ContinuousMulAction G α] variable [MulAction G α] [LocallyDense G α] [ContinuousConstSMul G α]
-- TODO: clean this lemma to not mention W anymore? -- TODO: clean this lemma to not mention W anymore?
lemma proposition_3_2_subset lemma proposition_3_2_subset

5
Rubin/RigidStabilizerBasis.lean
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@ -113,7 +113,7 @@ by
unfold asSets RigidStabilizerBasis unfold asSets RigidStabilizerBasis
simp simp
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
lemma RigidStabilizerBasis.smulImage_mem₀ {H : Subgroup G} (H_in_ristBasis : H ∈ RigidStabilizerBasis G α) lemma RigidStabilizerBasis.smulImage_mem₀ {H : Subgroup G} (H_in_ristBasis : H ∈ RigidStabilizerBasis G α)
(f : G) : ((fun g => f * g * f⁻¹) '' H.carrier) ∈ RigidStabilizerBasis.asSets G α := (f : G) : ((fun g => f * g * f⁻¹) '' H.carrier) ∈ RigidStabilizerBasis.asSets G α :=
@ -183,8 +183,7 @@ by
exact y_in_H exact y_in_H
def RigidStabilizerBasisMul (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α] def RigidStabilizerBasisMul (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α]
[ContinuousMulAction G α] (f : G) (H : Subgroup G) [ContinuousConstSMul G α] (f : G) (H : Subgroup G) : Subgroup G
: Subgroup G
where where
carrier := (fun g => f * g * f⁻¹) '' H.carrier carrier := (fun g => f * g * f⁻¹) '' H.carrier
mul_mem' := by mul_mem' := by

59
Rubin/SmulImage.lean
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@ -341,24 +341,15 @@ by
exact h_notin_V exact h_notin_V
#align distinct_images_from_disjoint Rubin.smulImage_distinct_of_disjoint_pow #align distinct_images_from_disjoint Rubin.smulImage_distinct_of_disjoint_pow
theorem continuousMulAction_elem_continuous {G : Type _} (α : Type _)
[Group G] [TopologicalSpace α] [MulAction G α] [hc : ContinuousMulAction G α] (g : G):
∀ (S : Set α), IsOpen S → IsOpen (g •'' S) ∧ IsOpen ((g⁻¹) •'' S) :=
by
intro S S_open
repeat rw [smulImage_eq_inv_preimage]
rw [inv_inv]
constructor
· exact (hc.continuous g⁻¹).isOpen_preimage _ S_open
· exact (hc.continuous g).isOpen_preimage _ S_open
theorem smulImage_isOpen {G α : Type _} theorem smulImage_isOpen {G α : Type _}
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] (g : G) [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] (g : G)
{S : Set α} (S_open : IsOpen S) : IsOpen (g •'' S) := {S : Set α} (S_open : IsOpen S) : IsOpen (g •'' S) :=
(continuousMulAction_elem_continuous α g S S_open).left by
rw [smulImage_eq_inv_preimage]
exact (continuous_id.const_smul g⁻¹).isOpen_preimage S S_open
theorem smulImage_isClosed {G α : Type _} theorem smulImage_isClosed {G α : Type _}
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] (g : G) [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] (g : G)
{S : Set α} (S_open : IsClosed S) : IsClosed (g •'' S) := {S : Set α} (S_open : IsClosed S) : IsClosed (g •'' S) :=
by by
rw [<-isOpen_compl_iff] rw [<-isOpen_compl_iff]
@ -367,9 +358,8 @@ by
apply smulImage_isOpen apply smulImage_isOpen
assumption assumption
theorem smulImage_interior' {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] theorem smulImage_interior {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
(g : G) (U : Set α) [hc : ContinuousConstSMul G α] (g : G) (U : Set α) :
(g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)):
interior (g •'' U) = g •'' interior U := interior (g •'' U) = g •'' interior U :=
by by
unfold interior unfold interior
@ -381,7 +371,7 @@ by
· intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩ · intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩
use g⁻¹ •'' T use g⁻¹ •'' T
repeat' apply And.intro repeat' apply And.intro
· exact (g_continuous T T_open).right · exact smulImage_isOpen g⁻¹ T_open
· rw [smulImage_subset_inv] · rw [smulImage_subset_inv]
rw [inv_inv] rw [inv_inv]
exact T_sub exact T_sub
@ -390,27 +380,15 @@ by
· intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩ · intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩
use g •'' T use g •'' T
repeat' apply And.intro repeat' apply And.intro
· exact (g_continuous T T_open).left · exact smulImage_isOpen g T_open
· apply smulImage_mono · apply smulImage_mono
exact T_sub exact T_sub
· exact x_in_T · exact x_in_T
theorem smulImage_interior {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] theorem smulImage_closure {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
[ContinuousMulAction G α] (g : G) (U : Set α) : [ContinuousConstSMul G α] (g : G) (U : Set α) :
interior (g •'' U) = g •'' interior U :=
smulImage_interior' g U (continuousMulAction_elem_continuous α g)
theorem smulImage_closure' {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
(g : G) (U : Set α)
(g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)):
closure (g •'' U) = g •'' closure U := closure (g •'' U) = g •'' closure U :=
by by
have g_continuous' : ∀ S : Set α, IsClosed S → IsClosed (g •'' S) ∧ IsClosed (g⁻¹ •'' S) := by
intro S S_closed
rw [<-isOpen_compl_iff] at S_closed
repeat rw [<-isOpen_compl_iff]
repeat rw [smulImage_compl]
exact g_continuous _ S_closed
unfold closure unfold closure
rw [smulImage_sInter] rw [smulImage_sInter]
simp simp
@ -421,21 +399,16 @@ by
rw [<-T'_eq] rw [<-T'_eq]
clear T' T'_eq clear T' T'_eq
apply IH apply IH
· exact (g_continuous' _ T_closed).left · exact smulImage_isClosed g T_closed
· apply smulImage_mono · apply smulImage_mono
exact U_ss_T exact U_ss_T
· intro IH T T_closed gU_ss_T · intro IH T T_closed gU_ss_T
apply IH apply IH
· exact (g_continuous' _ T_closed).right · exact smulImage_isClosed g⁻¹ T_closed
· rw [<-smulImage_subset_inv] · rw [<-smulImage_subset_inv]
exact gU_ss_T exact gU_ss_T
· simp · simp
theorem smulImage_closure {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
[ContinuousMulAction G α] (g : G) (U : Set α) :
closure (g •'' U) = g •'' closure U :=
smulImage_closure' g U (continuousMulAction_elem_continuous α g)
section Filters section Filters
open Topology open Topology
@ -521,7 +494,7 @@ by
variable [TopologicalSpace α] variable [TopologicalSpace α]
theorem smulFilter_nhds (g : G) (p : α) [ContinuousMulAction G α]: theorem smulFilter_nhds (g : G) (p : α) [ContinuousConstSMul G α]:
g •ᶠ 𝓝 p = 𝓝 (g • p) := g •ᶠ 𝓝 p = 𝓝 (g • p) :=
by by
ext S ext S
@ -545,7 +518,7 @@ by
· rw [mem_smulImage, inv_inv] · rw [mem_smulImage, inv_inv]
assumption assumption
theorem smulFilter_clusterPt (g : G) (F : Filter α) (x : α) [ContinuousMulAction G α] : theorem smulFilter_clusterPt (g : G) (F : Filter α) (x : α) [ContinuousConstSMul G α] :
ClusterPt x (g •ᶠ F) ↔ ClusterPt (g⁻¹ • x) F := ClusterPt x (g •ᶠ F) ↔ ClusterPt (g⁻¹ • x) F :=
by by
suffices ∀ (g : G) (F : Filter α) (x : α), ClusterPt x (g •ᶠ F) → ClusterPt (g⁻¹ • x) F by suffices ∀ (g : G) (F : Filter α) (x : α), ClusterPt x (g •ᶠ F) → ClusterPt (g⁻¹ • x) F by
@ -571,7 +544,7 @@ by
rw [inv_inv, smulImage_mul, mul_left_inv, one_smulImage] rw [inv_inv, smulImage_mul, mul_left_inv, one_smulImage]
assumption assumption
theorem smulImage_compact [ContinuousMulAction G α] (g : G) {U : Set α} (U_compact : IsCompact U) : theorem smulImage_compact [ContinuousConstSMul G α] (g : G) {U : Set α} (U_compact : IsCompact U) :
IsCompact (g •'' U) := IsCompact (g •'' U) :=
by by
intro F F_neBot F_le_principal intro F F_neBot F_le_principal

16
Rubin/Support.lean
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@ -3,6 +3,7 @@ import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Algebra.ConstMulAction
import Rubin.MulActionExt import Rubin.MulActionExt
import Rubin.SmulImage import Rubin.SmulImage
@ -381,17 +382,9 @@ variable {G α : Type _}
variable [Group G] variable [Group G]
variable [TopologicalSpace α] variable [TopologicalSpace α]
variable [MulAction G α] variable [MulAction G α]
variable [ContinuousMulAction G α] variable [ContinuousConstSMul G α]
theorem img_open_open (g : G) (U : Set α) (h : IsOpen U): IsOpen (g •'' U) := theorem support_isOpen (g : G) [T2Space α]: IsOpen (Support α g) := by
by
rw [Rubin.smulImage_eq_inv_preimage]
exact Continuous.isOpen_preimage (Rubin.ContinuousMulAction.continuous g⁻¹) U h
#align img_open_open Rubin.img_open_open
theorem support_open (g : G) [T2Space α]: IsOpen (Support α g) :=
by
apply isOpen_iff_forall_mem_open.mpr apply isOpen_iff_forall_mem_open.mpr
intro x xmoved intro x xmoved
rcases T2Space.t2 (g • x) x xmoved with ⟨U, V, open_U, open_V, gx_in_U, x_in_V, disjoint_U_V⟩ rcases T2Space.t2 (g • x) x xmoved with ⟨U, V, open_U, open_V, gx_in_U, x_in_V, disjoint_U_V⟩
@ -401,10 +394,9 @@ theorem support_open (g : G) [T2Space α]: IsOpen (Support α g) :=
disjoint_U_V disjoint_U_V
(mem_inv_smulImage.mp (Set.mem_of_mem_inter_right yW)) (mem_inv_smulImage.mp (Set.mem_of_mem_inter_right yW))
(Set.mem_of_mem_inter_left yW), (Set.mem_of_mem_inter_left yW),
IsOpen.inter open_V (Rubin.img_open_open g⁻¹ U open_U), IsOpen.inter open_V (smulImage_isOpen g⁻¹ open_U),
⟨x_in_V, mem_inv_smulImage.mpr gx_in_U⟩ ⟨x_in_V, mem_inv_smulImage.mpr gx_in_U⟩
⟩ ⟩
#align support_open Rubin.support_open
end Continuous end Continuous

37
Rubin/Topology.lean
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@ -2,48 +2,13 @@ import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Topology.Basic import Mathlib.Topology.Basic
import Mathlib.Topology.Homeomorph import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Algebra.ConstMulAction
import Mathlib.Data.Set.Basic import Mathlib.Data.Set.Basic
import Rubin.MulActionExt import Rubin.MulActionExt
namespace Rubin namespace Rubin
/--
Specificies that a group action is continuous, that is, for every group element `g`, `x ↦ g • x` is continuous.
Note that this is a weaker statement than `ContinuousSMul`, as the group `G` is not required to be a topology.
--/
class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] where
continuous : ∀ g : G, Continuous (fun x: α => g • x)
#align continuous_mul_action Rubin.ContinuousMulAction
def ContinuousMulAction.toHomeomorph {G : Type _} (α : Type _)
[Group G] [TopologicalSpace α] [MulAction G α] [hc : ContinuousMulAction G α]
(g : G) : Homeomorph α α
where
toFun := fun x => g • x
invFun := fun x => g⁻¹ • x
left_inv := by
intro y
simp
right_inv := by
intro y
simp
continuous_toFun := by
simp
exact hc.continuous _
continuous_invFun := by
simp
exact hc.continuous _
theorem ContinuousMulAction.toHomeomorph_toFun {G : Type _} (α : Type _)
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
(g : G) : (ContinuousMulAction.toHomeomorph α g).toFun = fun x => g • x := rfl
theorem ContinuousMulAction.toHomeomorph_invFun {G : Type _} (α : Type _)
[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
(g : G) : (ContinuousMulAction.toHomeomorph α g).invFun = fun x => g⁻¹ • x := rfl
-- TODO: give this a notation? -- TODO: give this a notation?
-- TODO: coe to / extend MulActionHom -- TODO: coe to / extend MulActionHom
structure EquivariantHomeomorph (G α β : Type _) [Group G] [TopologicalSpace α] structure EquivariantHomeomorph (G α β : Type _) [Group G] [TopologicalSpace α]

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