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8e4d533 | 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 | import Mathlib
namespace OAI
section
section
section
noncomputable section
namespace CannonVertexLinks
open Set Metric Topology
lemma finite_closed_local {J X : Type*} [Finite J] [MetricSpace X]
(C : J → Set X) (hC : ∀ j,IsClosed (C j)) (q : X) :
∃ r>0,∀ z∈ball q r,∀ j,z∈C j → q∈C j := by
classical
let U := ⋃ j : {j : J | q∉C j},C j.val
have hU : IsClosed U := isClosed_iUnion_of_finite fun j=>hC j.val
have hq : q∈Uᶜ := by
rintro ⟨S,⟨j,rfl⟩,hq⟩
exact j.property hq
obtain ⟨r,hr,hsub⟩ := Metric.isOpen_iff.mp hU.isOpen_compl q hq
refine ⟨r,hr,?_⟩
intro z hz j hj
by_contra hn
exact hsub hz (mem_iUnion.mpr ⟨⟨j,hn⟩,hj⟩)
lemma finite_pos_lower {J : Type*} [Finite J] (f : J → ℝ) (hf : ∀ j,0<f j) :
∃ r>0,∀ j,r<f j := by
have ho : IsOpen (⋂j,Iio (f j)) := isOpen_iInter_of_finite fun _=>isOpen_Iio
have hzero : (0:ℝ)∈⋂j,Iio (f j) := mem_iInter.mpr hf
obtain ⟨ε,hε,hsub⟩ := Metric.isOpen_iff.mp ho 0 hzero
refine ⟨ε/2,half_pos hε,mem_iInter.mp (hsub ?_)⟩
rw [mem_ball,dist_zero_right,Real.norm_eq_abs,abs_of_pos (half_pos hε)]
linarith
end CannonVertexLinks
end
end
end
end
end OAI
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