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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 Rubin.MulActionExt
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import Rubin.SmulImage
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import Rubin.Support
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namespace Rubin.Topological
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class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends
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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.Topological.ContinuousMulAction
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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.Topological.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.Topological.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.Topological.equivariant_inv
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variable [Rubin.Topological.ContinuousMulAction G α]
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theorem img_open_open (g : G) (U : Set α) (h : IsOpen U): IsOpen (g •'' U) :=
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by
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rw [Rubin.smulImage_eq_inv_preimage]
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exact Continuous.isOpen_preimage (Rubin.Topological.ContinuousMulAction.continuous g⁻¹) U h
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#align img_open_open Rubin.Topological.img_open_open
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theorem support_open (g : G) [TopologicalSpace α] [T2Space α]
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[Rubin.Topological.ContinuousMulAction G α] : IsOpen (Support α g) :=
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by
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apply isOpen_iff_forall_mem_open.mpr
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intro x xmoved
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rcases T2Space.t2 (g • x) x xmoved with ⟨U, V, open_U, open_V, gx_in_U, x_in_V, disjoint_U_V⟩
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exact
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⟨V ∩ (g⁻¹ •'' U), fun y yW =>
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Disjoint.ne_of_mem
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disjoint_U_V
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(mem_inv_smulImage.mp (Set.mem_of_mem_inter_right yW))
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(Set.mem_of_mem_inter_left yW),
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IsOpen.inter open_V (Rubin.Topological.img_open_open g⁻¹ U open_U),
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⟨x_in_V, mem_inv_smulImage.mpr gx_in_U⟩
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⟩
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#align support_open Rubin.Topological.support_open
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end Rubin.Topological
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