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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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import Mathlib
namespace OAI
section
section
section
noncomputable section
namespace CannonLocalRealization
open Set Metric Topology
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]
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
end
end
end
end
end OAI
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