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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 | 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
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