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import Mathlib.GroupTheory.Subgroup.Basic
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import Mathlib.GroupTheory.GroupAction.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Topology.Homeomorph
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import Mathlib.Data.Set.Basic
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import Rubin.MulActionExt
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namespace Rubin
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class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] where
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continuous : ∀ g : G, Continuous (fun x: α => g • x)
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#align continuous_mul_action Rubin.ContinuousMulAction
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def ContinuousMulAction.toHomeomorph {G : Type _} (α : Type _)
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[Group G] [TopologicalSpace α] [MulAction G α] [hc : ContinuousMulAction G α]
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(g : G) : Homeomorph α α
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where
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toFun := fun x => g • x
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invFun := fun x => g⁻¹ • x
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left_inv := by
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intro y
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simp
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right_inv := by
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intro y
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simp
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continuous_toFun := by
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simp
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exact hc.continuous _
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continuous_invFun := by
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simp
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exact hc.continuous _
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theorem ContinuousMulAction.toHomeomorph_toFun {G : Type _} (α : Type _)
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[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
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(g : G) : (ContinuousMulAction.toHomeomorph α g).toFun = fun x => g • x := rfl
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theorem ContinuousMulAction.toHomeomorph_invFun {G : Type _} (α : Type _)
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[Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
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(g : G) : (ContinuousMulAction.toHomeomorph α g).invFun = fun x => g⁻¹ • x := rfl
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-- TODO: give this a notation?
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structure EquivariantHomeomorph (G α β : Type _) [Group G] [TopologicalSpace α]
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[TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where
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equivariant : is_equivariant G toFun
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#align equivariant_homeomorph Rubin.EquivariantHomeomorph
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variable {G α β : Type _}
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variable [Group G]
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variable [TopologicalSpace α] [TopologicalSpace β]
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theorem equivariant_fun [MulAction G α] [MulAction G β]
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(h : EquivariantHomeomorph G α β) :
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is_equivariant G h.toFun :=
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h.equivariant
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#align equivariant_fun Rubin.equivariant_fun
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theorem equivariant_inv [MulAction G α] [MulAction G β]
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(h : EquivariantHomeomorph G α β) :
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is_equivariant G h.invFun :=
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by
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intro g x
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symm
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let e := congr_arg h.invFun (h.equivariant g (h.invFun x))
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rw [h.left_inv _, h.right_inv _] at e
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exact e
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#align equivariant_inv Rubin.equivariant_inv
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open Topology
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/--
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Note: `𝓝[≠] x` is notation for `nhdsWithin x {[x]}ᶜ`, ie. the neighborhood of x not containing itself.
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--/
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class HasNoIsolatedPoints (α : Type _) [TopologicalSpace α] :=
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nhbd_ne_bot : ∀ x : α, 𝓝[≠] x ≠ ⊥
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#align has_no_isolated_points Rubin.HasNoIsolatedPoints
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instance has_no_isolated_points_neBot {α : Type _} [TopologicalSpace α] [h_nip: HasNoIsolatedPoints α] (x: α): Filter.NeBot (𝓝[≠] x) where
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ne' := h_nip.nhbd_ne_bot x
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end Rubin
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