| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function Metric MeasureTheory | |
| open scoped Topology ENNReal | |
| open Set Filter Function Metric Manifold | |
| open scoped Topology ContDiff | |
| namespace YauCounterexamples | |
| variable {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] [Fintype ι] | |
| theorem compact_disjoint_smooth_cutoffs (K V : ι → Set E) | |
| (hK : ∀ i, IsCompact (K i)) (hd : Pairwise (Disjoint on K)) | |
| (hV : ∀ i, IsOpen (V i)) (hKV : ∀ i, K i ⊆ V i) : | |
| ∃ χ : ι → E → ℝ, | |
| (∀ i, ContDiff ℝ ∞ (χ i) ∧ HasCompactSupport (χ i) ∧ | |
| tsupport (χ i) ⊆ V i ∧ (∀ᶠ x in 𝓝ˢ (K i), χ i x = 1) ∧ | |
| ∀ x, χ i x ∈ Icc 0 1) ∧ Pairwise (Disjoint on fun i => tsupport (χ i)) := by | |
| have hN : Pairwise (Disjoint on fun i => 𝓝ˢ (K i)) := by | |
| intro i j hij | |
| change Disjoint (𝓝ˢ (K i)) (𝓝ˢ (K j)) | |
| rw [(hK i).disjoint_nhdsSet_left] | |
| intro x hx | |
| rw [(hK j).disjoint_nhdsSet_right] | |
| intro y hy | |
| exact disjoint_nhds_nhds.mpr (fun hxy => (hd hij).notMem_of_mem_left hx (hxy ▸ hy)) | |
| obtain ⟨O,hO,hdO⟩ := hN.exists_mem_filter_basis_of_disjoint (fun i => hasBasis_nhdsSet (K i)) | |
| have hex (i : ι) : ∃ χ : E → ℝ, ContDiff ℝ ∞ χ ∧ HasCompactSupport χ ∧ | |
| tsupport χ ⊆ V i ∩ O i ∧ (∀ᶠ x in 𝓝ˢ (K i), χ x = 1) ∧ ∀ x, χ x ∈ Icc 0 1 := by | |
| obtain ⟨B,hB,hKB,hBc⟩ := exists_isOpen_superset_and_isCompact_closure (hK i) | |
| have hNV : V i ∩ O i ∩ B ∈ 𝓝ˢ (K i) := | |
| ((hV i).inter (hO i).1 |>.inter hB).mem_nhdsSet.mpr | |
| (fun x hx => ⟨⟨hKV i hx,(hO i).2 hx⟩,hKB hx⟩) | |
| obtain ⟨W,hW,hKW,hWsub⟩ := (hK i).exists_isOpen_closure_subset hNV | |
| obtain ⟨f,hf1,hf0,hf⟩ := exists_contMDiffMap_one_nhds_of_subset_interior | |
| (I := 𝓘(ℝ,E)) (n := ⊤) (hK i).isClosed (t := W) (by simpa only [hW.interior_eq] using hKW) | |
| have hfs : tsupport (f : E → ℝ) ⊆ closure W := by | |
| apply closure_mono | |
| intro x hx | |
| by_contra hh | |
| exact hx (hf0 x hh) | |
| refine ⟨f,contMDiff_iff_contDiff.mp f.contMDiff,?_,?_,hf1,hf⟩ | |
| · exact hBc.of_isClosed_subset (isClosed_tsupport _) | |
| (hfs.trans (hWsub.trans (fun x hx => subset_closure hx.2))) | |
| · exact hfs.trans (hWsub.trans (fun x hx => hx.1)) | |
| choose χ hχ using hex | |
| refine ⟨χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_left, | |
| (hχ i).2.2.2.1,(hχ i).2.2.2.2⟩,?_⟩ | |
| intro i j hij | |
| exact (hdO hij).mono ((hχ i).2.2.1.trans inter_subset_right) | |
| ((hχ j).2.2.1.trans inter_subset_right) | |
| end YauCounterexamples | |
| end | |
| end OAI | |