| 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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|
