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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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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
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open Set unitInterval Topology
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open Set Topology unitInterval
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open Set Metric Topology unitInterval
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open Set Metric Topology
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open Set unitInterval
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open Set Metric Topology
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open Set Metric Topology
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open Set Metric Topology
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open Set Topology
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open Set Topology
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open Set Topology
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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
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