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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A] [TopologicalSpace A] [IsTopologicalGroup A]
lemma isOpen_of_isClosed_relIndex (K H : Subgroup A) (hK : IsOpen (K:Set A))
(hH : IsClosed (H:Set A)) (hi : H.relIndex K≠0) : IsOpen (H:Set A) := by
let J := H.subgroupOf K
let : J.FiniteIndex := ⟨hi⟩
have hJ : IsOpen (J:Set K) := J.isOpen_of_isClosed_of_finiteIndex
(hH.preimage continuous_subtype_val)
have hI : IsOpen ((J.map K.subtype):Set A) := hK.isOpenMap_subtype_val _ hJ
exact Subgroup.isOpen_mono (by rintro _ ⟨k,hk,rfl⟩; exact hk) hI
lemma topologicalClosure_open_of_relIndex (K N : Subgroup A) (hK : IsOpen (K:Set A))
(hi : N.relIndex K≠0) : IsOpen (N.topologicalClosure:Set A) := by
apply isOpen_of_isClosed_relIndex K N.topologicalClosure hK N.isClosed_topologicalClosure
exact fun h => hi (Subgroup.relIndex_eq_zero_of_le_left N.le_topologicalClosure h)
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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