| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| namespace CannonFacialStars | |
| open Set Topology | |
| variable {X D V : Type*} [TopologicalSpace X] [Finite D] | |
| theorem interior_union_at (C : D → Set X) (hC : ∀ d,IsClosed (C d)) | |
| (hcover : ∀ x,∃ d,x∈C d) (S : Set D) {x : X} | |
| (hx : ∀ d,x∈C d → d∈S) : x∈interior (⋃ d∈S,C d) := by | |
| let B := ⋃ d∈Sᶜ,C d | |
| have hB : IsClosed B := isClosed_iUnion_of_finite fun d => | |
| isClosed_iUnion_of_finite fun _ => hC d | |
| have hxB : x∈Bᶜ := by | |
| intro hxB | |
| obtain ⟨d,hd,hxd⟩ := mem_iUnion₂.mp hxB | |
| exact hd (hx d hxd) | |
| apply mem_interior.mpr | |
| refine ⟨Bᶜ,?_,hB.isOpen_compl,hxB⟩ | |
| intro y hy | |
| obtain ⟨d,hd⟩ := hcover y | |
| have hdS : d∈S := by | |
| by_contra hh | |
| exact hy (mem_iUnion₂.mpr ⟨d,hh,hd⟩) | |
| exact mem_iUnion₂.mpr ⟨d,hdS,hd⟩ | |
| end CannonFacialStars | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |