| import OAI.Analysis.CircleDomains.Topology.PhysicalPlaneCoordinates |
| import Mathlib.Topology.Separation.Regular |
|
|
| namespace OAI |
|
|
| /-! The compact exterior and relatively closed plateau sets used when |
| smoothing the filled first-arrival barrier. -/ |
|
|
| open Set Function Filter |
| open scoped Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem exists_compact_plane_exterior_of_nhds |
| {V : Set Sphere} (hV : V ∈ 𝓝 OnePoint.infty) : |
| ∃K : Set EuclideanPlane, IsCompact K ∧ ∀x ∉ K, planeSphere x ∈ V := by |
| obtain ⟨S, ⟨_, hS⟩, hSV⟩ := OnePoint.hasBasis_nhds_infty.mem_iff.mp hV |
| refine ⟨complexPlaneCoordinates.symm '' S, hS.image complexPlaneCoordinates.symm.continuous, ?_⟩ |
| intro x hx |
| apply hSV |
| left |
| refine ⟨complexPlaneCoordinates x, ?_, rfl⟩ |
| intro hz |
| exact hx ⟨complexPlaneCoordinates x, hz, complexPlaneCoordinates.symm_apply_apply x⟩ |
|
|
| theorem physical_plateau_relative_complement_open |
| {X : Type*} [TopologicalSpace X] {G : Set Sphere} (hG : IsOpen G) |
| {π : Sphere → X} (hπ : Continuous π) {A : Set X} (hA : IsClosed A) : |
| IsOpen (physicalPlaneDomain G \ |
| (physicalPlaneDomain G ∩ (π ∘ planeSphere) ⁻¹' A)) := by |
| have heq : physicalPlaneDomain G \ |
| (physicalPlaneDomain G ∩ (π ∘ planeSphere) ⁻¹' A) = |
| physicalPlaneDomain G ∩ (π ∘ planeSphere) ⁻¹' Aᶜ := by |
| ext x |
| simp only [Set.mem_sdiff, mem_inter_iff, mem_preimage, mem_compl_iff] |
| tauto |
| rw [heq] |
| exact (isOpen_physicalPlaneDomain hG).inter (hA.isOpen_compl.preimage |
| (hπ.comp continuous_planeSphere)) |
|
|
| theorem exists_closed_anchor_plateau |
| {X : Type*} [TopologicalSpace X] [RegularSpace X] |
| {E F : Set X} (hF : IsCompact F) (hFE : F ⊆ interior E) : |
| ∃A V : Set X, IsClosed A ∧ IsOpen V ∧ F ⊆ interior A ∧ A ⊆ V ∧ V ⊆ E := by |
| obtain ⟨U, hU, hFU, hUE⟩ := hF.exists_isOpen_closure_subset |
| (isOpen_interior.mem_nhdsSet.mpr hFE) |
| refine ⟨closure U, interior E, isClosed_closure, isOpen_interior, ?_, hUE, interior_subset⟩ |
| exact hFU.trans (hU.subset_interior_closure) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
