| import Mathlib |
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| namespace OAI |
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| namespace BorsukNine |
| open Set |
| open scoped Topology |
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| lemma compact_local_homotopy {X E : Type*} [TopologicalSpace X] [T2Space X] |
| [LocallyCompactSpace X] [TopologicalSpace E] [Zero E] |
| {U C : Set X} (hU : IsOpen U) (hC : IsCompact C) (hCU : C ⊆ U) |
| (H : X → E) (hH : Continuous H) (hHz : ∀ x, H x ≠ 0) |
| (K : ℝ × X → E) (hK : ContinuousOn K (Icc (0:ℝ) 1 ×ˢ U)) |
| (hK0 : ∀ x ∈ U, K (0,x) = H x) |
| (hKz : ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ U, K (t,x) ≠ 0) |
| (B : Set E) (houtside : ∀ x ∉ U, H x ∉ B) |
| (hhits : ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ U, K (t,x) ∈ B → x ∈ C) |
| (hend : ∀ x ∈ U, K (1,x) ∉ B) : |
| ∃ G : ℝ × X → E, ContinuousOn G (Icc (0:ℝ) 1 ×ˢ Set.univ) ∧ |
| (∀ x, G (0,x) = H x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, G (t,x) ≠ 0) ∧ |
| (∀ x, G (1,x) ∉ B) := by |
| classical |
| obtain ⟨q,hq1,-,hqs,hqb⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hC hU hCU |
| let G : ℝ × X → E := fun p => if p.2 ∈ U then K (p.1 * q p.2,p.2) else H p.2 |
| have htm (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (x : X) : t * q x ∈ Icc (0:ℝ) 1 := |
| ⟨mul_nonneg ht.1 (hqb x).1, |
| (mul_le_mul_of_nonneg_right ht.2 (hqb x).1).trans (by simpa using (hqb x).2)⟩ |
| have hG : ContinuousOn G (Icc (0:ℝ) 1 ×ˢ Set.univ) := by |
| intro p hp |
| by_cases hx : p.2 ∈ U |
| · have hc : ContinuousOn (fun p : ℝ × X => K (p.1 * q p.2,p.2)) |
| (Icc (0:ℝ) 1 ×ˢ U) := hK.comp |
| ((continuous_fst.mul (q.continuous.comp continuous_snd)).prodMk continuous_snd).continuousOn |
| (fun p hp => ⟨htm p.1 hp.1 p.2,hp.2⟩) |
| have hnear : ∀ᶠ p' in 𝓝 p, p'.2 ∈ U := |
| continuous_snd.continuousAt.preimage_mem_nhds (hU.mem_nhds hx) |
| have hset : Icc (0:ℝ) 1 ×ˢ U ∈ 𝓝[Icc (0:ℝ) 1 ×ˢ Set.univ] p := by |
| filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear, |
| self_mem_nhdsWithin] with p' hu hi |
| exact ⟨hi.1,hu⟩ |
| apply ((hc p ⟨hp.1,hx⟩).mono_of_mem_nhdsWithin hset).congr_of_eventuallyEq |
| · filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear] with p' hu |
| simp [G,hu] |
| · simp [G,hx] |
| · have hxq : p.2 ∉ tsupport q := fun h => hx (hqs h) |
| have hnear : ∀ᶠ p' in 𝓝 p, p'.2 ∉ tsupport q := |
| continuous_snd.continuousAt.preimage_mem_nhds ((isClosed_tsupport q).isOpen_compl.mem_nhds hxq) |
| apply (hH.comp continuous_snd).continuousAt.continuousWithinAt.congr_of_eventuallyEq |
| · filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hnear] with p' hq |
| have hz : q p'.2 = 0 := image_eq_zero_of_notMem_tsupport hq |
| by_cases hu : p'.2 ∈ U |
| · simp [G,hu,hz,hK0 p'.2 hu] |
| · simp [G,hu] |
| · simp [G,hx] |
| refine ⟨G,hG,?_,?_,?_⟩ |
| · intro x |
| by_cases hx : x ∈ U |
| · simp [G,hx,hK0 x hx] |
| · simp [G,hx] |
| · intro t ht x |
| by_cases hx : x ∈ U |
| · simpa only [G,ite_eq_left hx] using hKz _ (htm t ht x) x hx |
| · simpa only [G,ite_eq_right hx] using hHz x |
| · intro x hb |
| by_cases hx : x ∈ U |
| · have hh : K (q x,x) ∈ B := by simpa only [G,ite_eq_left hx,one_mul] using hb |
| have hc := hhits _ (hqb x) x hx hh |
| rw [hq1 hc] at hh |
| exact hend x hx hh |
| · exact houtside x hx (by simpa only [G,ite_eq_right hx] using hb) |
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| end BorsukNine |
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| end OAI |
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