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rubin-lean4/Rubin/Topological.lean

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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
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
theorem continuousMulAction_elem_continuous {G : Type _} (α : Type _)
[Group G] [TopologicalSpace 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
-- 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