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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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.CompactQuotientProper
namespace OAI
section
noncomputable section
open Set Function Topology Filter
namespace CriticalPercolation.CompactQuotient
variable {A : Type*} [Group A] [TopologicalSpace A] [IsTopologicalGroup A] [T2Space A]
lemma closed_of_locallyCompact_subgroup (L : Subgroup A) [LocallyCompactSpace L] :
IsClosed (L : Set A) := by
let _ : UniformSpace A := IsTopologicalGroup.rightUniformSpace A
have hright : IsRightUniformGroup L := by
refine {uniformity_eq := ?_}
change Filter.comap (fun p : L × L => ((p.1 : A),(p.2 : A)))
(Filter.comap (fun p : A × A => p.2*p.1⁻¹) (𝓝 (1 : A))) =
Filter.comap (fun p : L × L => p.2*p.1⁻¹) (𝓝 (1 : L))
rw [nhds_subtype_eq_comap,Filter.comap_comap,Filter.comap_comap]
rfl
let _ := hright
let _ : CompleteSpace L := IsRightUniformGroup.completeSpace_of_weaklyLocallyCompactSpace L
exact (completeSpace_coe_iff_isComplete.mp (inferInstance : CompleteSpace L)).isClosed
lemma closed_of_compact_kernel_locallyCompact_quotient
(L : Subgroup A) (H : Subgroup L) [H.Normal]
(hH : IsCompact (H : Set L)) [LocallyCompactSpace (L ⧸ H)] :
IsClosed (L : Set A) := by
let _ : LocallyCompactSpace L := locallyCompact_of_compact_kernel H hH
exact closed_of_locallyCompact_subgroup L
lemma open_of_finiteIndex_compact_kernel_locallyCompact_quotient
(L : Subgroup A) [L.FiniteIndex] (H : Subgroup L) [H.Normal]
(hH : IsCompact (H : Set L)) [LocallyCompactSpace (L ⧸ H)] :
IsOpen (L : Set A) :=
L.isOpen_of_isClosed_of_finiteIndex
(closed_of_compact_kernel_locallyCompact_quotient L H hH)
end CriticalPercolation.CompactQuotient
end
end
end OAI
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