| import Mathlib |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| namespace CannonLocalRealization |
| open Set Metric Topology |
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| theorem exists_local_compact_surjection (K U : Set ℂ) (hK : IsCompact K) |
| (hU : IsOpen U) (hn : U.Nonempty) (g : C(ℂ,ℂ)) : |
| ∃ f : C(ℂ,ℂ),EqOn f g Uᶜ ∧ |
| ∃ W : Set ℂ,IsCompact W ∧ W⊆U ∧ K⊆f '' W := by |
| obtain ⟨x₀,hx₀⟩:=hn |
| obtain ⟨r,hr,hball⟩:=Metric.isOpen_iff.mp hU x₀ hx₀ |
| obtain ⟨R,hR,hbound⟩:=hK.isBounded.exists_pos_norm_le |
| let s : ℝ := r/(2*R) |
| have hs : 0<s := by dsimp [s]; positivity |
| have hsR : s*R=r/2 := by dsimp [s]; field_simp |
| let E : ℂ → ℂ := fun y=>x₀+s • y |
| have he : Continuous E := continuous_const.add (continuous_id.const_smul s) |
| have hEU : E '' K⊆U := by |
| rintro _ ⟨y,hy,rfl⟩ |
| apply hball |
| rw [mem_ball,dist_eq_norm] |
| change ‖x₀+s • y-x₀‖<r |
| rw [add_sub_cancel_left,norm_smul,Real.norm_eq_abs,abs_of_pos hs] |
| exact (mul_le_mul_of_nonneg_left (hbound y hy) hs.le).trans_lt (by rw [hsR]; linarith) |
| obtain ⟨ψ,hψ,_,hψU,_⟩:=exists_continuousMap_one_of_isCompact_subset_isOpen (hK.image he) hU hEU |
| let A : ℂ → ℂ := fun x=>(s⁻¹:ℝ) • (x-x₀) |
| have hA : Continuous A := (continuous_id.sub continuous_const).const_smul (s⁻¹ : ℝ) |
| let f : C(ℂ,ℂ) := ⟨fun x=>(ψ x) • A x + (1-ψ x) • g x, |
| (ψ.continuous.smul hA).add ((continuous_const.sub ψ.continuous).smul g.continuous)⟩ |
| refine ⟨f,?_,?_⟩ |
| · intro x hx |
| have hz : ψ x=0 := by |
| by_contra hn |
| exact hx (hψU (subset_closure hn)) |
| change (ψ x) • A x + (1-ψ x) • g x=g x |
| simp [hz] |
| · refine ⟨E '' K,hK.image he,hEU,?_⟩ |
| intro y hy |
| refine ⟨E y,mem_image_of_mem E hy,?_⟩ |
| have h1 : ψ (E y)=1 := hψ (mem_image_of_mem E hy) |
| change (ψ (E y)) • A (E y) + (1-ψ (E y)) • g (E y)=y |
| simp only [h1,one_smul,sub_self,zero_smul,add_zero] |
| change (s⁻¹:ℝ) • (x₀+s • y-x₀)=y |
| rw [add_sub_cancel_left,smul_smul,inv_mul_cancel₀ hs.ne',one_smul] |
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| theorem exists_local_surjection (K U : Set ℂ) (hK : IsCompact K) |
| (hU : IsOpen U) (hn : U.Nonempty) (g : C(ℂ,ℂ)) : |
| ∃ f : C(ℂ,ℂ),EqOn f g Uᶜ ∧ K⊆f '' U := by |
| obtain ⟨f,hf,W,_,hWU,hW⟩:=exists_local_compact_surjection K U hK hU hn g |
| exact ⟨f,hf,hW.trans (image_mono hWU)⟩ |
| end CannonLocalRealization |
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| end |
| end |
| end |
| end |
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| end OAI |
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