| import Mathlib |
| import OAI.Geometry.Cannon.Energy.PuncturedBallConnected |
| import OAI.Geometry.Cannon.PlanarGraphs.RegionBoundaryHitsOpen |
| import OAI.Geometry.Cannon.PlanarGraphs.IncidentSubsetPatch |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology |
|
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| noncomputable section |
| noncomputable section |
| open Set unitInterval |
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| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
|
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| noncomputable section |
| open Set Metric Topology |
|
|
| noncomputable section |
| open Set Metric Topology |
|
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
|
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| noncomputable section |
| open Set Topology |
|
|
| noncomputable section |
| open Set Topology unitInterval |
|
|
| namespace CannonPlanarNeighborhood |
| open Set Metric Topology |
| open CannonFaceIncidence |
|
|
| theorem open_preconnected_remove_point {U : Set ℂ} (hU : IsOpen U) |
| (hc : IsPreconnected U) (x : ℂ) : IsPreconnected (U\{x}) := by |
| by_cases hx : x∈U |
| swap |
| · have he : U\{x}=U := by ext z; simp only [mem_sdiff,mem_singleton_iff]; constructor; exact And.left; intro hz; exact ⟨hz,fun h => hx (h ▸ hz)⟩ |
| rwa [he] |
| have hV : IsOpen (U\{x}) := hU.sdiff isClosed_singleton |
| let _ : LocallyConnectedSpace (U\{x} : Set ℂ) := hV.locallyConnectedSpace |
| let _ : DiscreteTopology (ConnectedComponents (U\{x} : Set ℂ)) := inferInstance |
| obtain ⟨r,hr,hrU⟩ := Metric.isOpen_iff.mp hU x hx |
| have hsub : ball x r\{x}⊆U\{x} := sdiff_subset_sdiff_left hrU |
| obtain ⟨d,hd⟩ := label_on_connected ConnectedComponents.continuous_coe |
| (punctured_ball_connected x hr) hsub |
| have hi : incident (U\{x}) ConnectedComponents.mk x⊆{d} := by |
| apply incident_subset_of_patch isOpen_ball _ (mem_ball_self hr) |
| intro y hy |
| exact hd ⟨hy.1,hy.2.2⟩ |
| have hconstant (e : ConnectedComponents (U\{x} : Set ℂ)) : e=d := by |
| obtain ⟨z,rfl⟩ := ConnectedComponents.surjective_coe e |
| obtain ⟨y,⟨hyU,hyV⟩,hy⟩ := region_boundary_hits_in_open hU hV sdiff_subset hc |
| ⟨x,hx,fun h => h.2 (mem_singleton x)⟩ ConnectedComponents.continuous_coe |
| (ConnectedComponents.mk z) ⟨z,z.property,rfl⟩ |
| have hyx : y=x := by by_contra hn; exact hyV ⟨hyU,hn⟩ |
| subst y |
| exact hi hy |
| have : PreconnectedSpace (U\{x} : Set ℂ) := by |
| rw [preconnectedSpace_iff_connectedComponent] |
| intro a |
| apply eq_univ_of_forall |
| intro b |
| exact ConnectedComponents.coe_eq_coe'.mp ((hconstant _).trans (hconstant _).symm) |
| exact isPreconnected_iff_preconnectedSpace.mpr inferInstance |
| |
| theorem open_preconnected_remove_finite {U : Set ℂ} (hU : IsOpen U) |
| (hc : IsPreconnected U) (T : Finset ℂ) : IsPreconnected (U\(T:Set ℂ)) := by |
| classical |
| induction T using Finset.induction_on with |
| | empty => simpa using hc |
| | @insert x T hx ih => |
| have he : U\(↑(insert x T):Set ℂ)=(U\(T:Set ℂ))\{x} := by ext z; simp; tauto |
| rw [he] |
| exact open_preconnected_remove_point (hU.sdiff T.finite_toSet.isClosed) ih x |
| end CannonPlanarNeighborhood |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
