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rubin-lean4/Rubin/Topology.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.MulActionExt
namespace Rubin
class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] [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
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
end Rubin