|
|
import Mathlib.GroupTheory.Subgroup.Basic
|
|
|
import Mathlib.GroupTheory.GroupAction.Basic
|
|
|
import Mathlib.Topology.Basic
|
|
|
import Mathlib.Topology.Homeomorph
|
|
|
import Mathlib.Data.Set.Basic
|
|
|
|
|
|
import Rubin.RigidStabilizer
|
|
|
import Rubin.MulActionExt
|
|
|
import Rubin.SmulImage
|
|
|
import Rubin.Support
|
|
|
|
|
|
namespace Rubin
|
|
|
|
|
|
section Continuity
|
|
|
|
|
|
-- TODO: don't have this extend MulAction
|
|
|
class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends
|
|
|
MulAction G α where
|
|
|
continuous : ∀ g : G, Continuous (fun x: α => g • x)
|
|
|
#align continuous_mul_action Rubin.ContinuousMulAction
|
|
|
|
|
|
-- TODO: give this a notation?
|
|
|
structure EquivariantHomeomorph (G α β : Type _) [Group G] [TopologicalSpace α]
|
|
|
[TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where
|
|
|
equivariant : is_equivariant G toFun
|
|
|
#align equivariant_homeomorph Rubin.EquivariantHomeomorph
|
|
|
|
|
|
variable {G α β : Type _}
|
|
|
variable [Group G]
|
|
|
variable [TopologicalSpace α] [TopologicalSpace β]
|
|
|
|
|
|
theorem equivariant_fun [MulAction G α] [MulAction G β]
|
|
|
(h : EquivariantHomeomorph G α β) :
|
|
|
is_equivariant G h.toFun :=
|
|
|
h.equivariant
|
|
|
#align equivariant_fun Rubin.equivariant_fun
|
|
|
|
|
|
theorem equivariant_inv [MulAction G α] [MulAction G β]
|
|
|
(h : EquivariantHomeomorph G α β) :
|
|
|
is_equivariant G h.invFun :=
|
|
|
by
|
|
|
intro g x
|
|
|
symm
|
|
|
let e := congr_arg h.invFun (h.equivariant g (h.invFun x))
|
|
|
rw [h.left_inv _, h.right_inv _] at e
|
|
|
exact e
|
|
|
#align equivariant_inv Rubin.equivariant_inv
|
|
|
|
|
|
variable [ContinuousMulAction G α]
|
|
|
|
|
|
theorem img_open_open (g : G) (U : Set α) (h : IsOpen U): IsOpen (g •'' U) :=
|
|
|
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) [TopologicalSpace α] [T2Space α]
|
|
|
[ContinuousMulAction G α] : IsOpen (Support α g) :=
|
|
|
by
|
|
|
apply isOpen_iff_forall_mem_open.mpr
|
|
|
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⟩
|
|
|
exact
|
|
|
⟨V ∩ (g⁻¹ •'' U), fun y yW =>
|
|
|
Disjoint.ne_of_mem
|
|
|
disjoint_U_V
|
|
|
(mem_inv_smulImage.mp (Set.mem_of_mem_inter_right yW))
|
|
|
(Set.mem_of_mem_inter_left yW),
|
|
|
IsOpen.inter open_V (Rubin.img_open_open g⁻¹ U open_U),
|
|
|
⟨x_in_V, mem_inv_smulImage.mpr gx_in_U⟩
|
|
|
⟩
|
|
|
#align support_open Rubin.support_open
|
|
|
|
|
|
end Continuity
|
|
|
|
|
|
-- TODO: come up with a name
|
|
|
section LocallyDense
|
|
|
open Topology
|
|
|
|
|
|
/--
|
|
|
Note: `𝓝[≠] x` is notation for `nhdsWithin x {[x]}ᶜ`, ie. the neighborhood of x not containing itself.
|
|
|
|
|
|
--/
|
|
|
class HasNoIsolatedPoints (α : Type _) [TopologicalSpace α] :=
|
|
|
nhbd_ne_bot : ∀ x : α, 𝓝[≠] x ≠ ⊥
|
|
|
#align has_no_isolated_points Rubin.HasNoIsolatedPoints
|
|
|
|
|
|
instance has_no_isolated_points_neBot {α : Type _} [TopologicalSpace α] [h_nip: HasNoIsolatedPoints α] (x: α): Filter.NeBot (𝓝[≠] x) where
|
|
|
ne' := h_nip.nhbd_ne_bot x
|
|
|
|
|
|
class LocallyDense (G α : Type _) [Group G] [TopologicalSpace α] extends MulAction G α :=
|
|
|
isLocallyDense:
|
|
|
∀ U : Set α,
|
|
|
∀ p ∈ U,
|
|
|
p ∈ interior (closure (MulAction.orbit (RigidStabilizer G U) p))
|
|
|
#align is_locally_dense Rubin.LocallyDense
|
|
|
|
|
|
lemma LocallyDense.nonEmpty {G α : Type _} [Group G] [TopologicalSpace α] [LocallyDense G α]:
|
|
|
∀ {U : Set α},
|
|
|
Set.Nonempty U →
|
|
|
∃ p ∈ U, p ∈ interior (closure (MulAction.orbit (RigidStabilizer G U) p)) :=
|
|
|
by
|
|
|
intros U H_ne
|
|
|
exact ⟨H_ne.some, H_ne.some_mem, LocallyDense.isLocallyDense U H_ne.some H_ne.some_mem⟩
|
|
|
|
|
|
end LocallyDense
|
|
|
|
|
|
|
|
|
end Rubin
|