import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.Topology.Basic import Mathlib.Topology.Homeomorph 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 -- 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 [Rubin.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] [Rubin.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 Other open Topology -- Note: `𝓝[≠] x` is notation for `nhdsWithin x {[x]}ᶜ`, ie. the neighborhood of x not containing itself -- TODO: make this a class? def has_no_isolated_points (α : Type _) [TopologicalSpace α] := ∀ x : α, 𝓝[≠] x ≠ ⊥ #align has_no_isolated_points Rubin.has_no_isolated_points instance has_no_isolated_points_neBot {α : Type _} [TopologicalSpace α] (h_nip: has_no_isolated_points α) (x: α): Filter.NeBot (𝓝[≠] x) where ne' := h_nip 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 namespace LocallyDense lemma 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 Other end Rubin