File size: 3,428 Bytes
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 | import Mathlib
namespace OAI
section
section
section
noncomputable section
namespace CannonFiniteCarrier
open Set Topology Relation
lemma closed_cover_reflTransGen {Y ι : Type*} [TopologicalSpace Y] [Finite ι]
(K : ι → Set Y) (hK : ∀ i,IsClosed (K i)) (S : Set Y)
(hS : IsPreconnected S) (hcover : S⊆⋃i,K i)
(i j : ι) (hi : (S∩K i).Nonempty) (hj : (S∩K j).Nonempty) :
ReflTransGen (fun a b=> (S∩K a∩K b).Nonempty) i j := by
classical
by_contra hij
let R := fun a b=> (S∩K a∩K b).Nonempty
let T : Set ι := {k | ReflTransGen R i k}
let U : Set Y := ⋃ k∈T,K k
let V : Set Y := ⋃ k∈Tᶜ,K k
have hsplit : (⋃ k,K k)=U∪V := iSup_split K (·∈T)
have hcU : IsClosed U := isClosed_iUnion_of_finite fun k=>isClosed_iUnion_of_finite fun _=>hK k
have hcV : IsClosed V := isClosed_iUnion_of_finite fun k=>isClosed_iUnion_of_finite fun _=>hK k
have hiT : i∈T := ReflTransGen.refl
obtain ⟨a,haS,hai⟩:=hi
obtain ⟨b,hbS,hbj⟩:=hj
have haU : (S∩U).Nonempty := ⟨a,haS,mem_iUnion₂_of_mem hiT hai⟩
have hbV : (S∩V).Nonempty := ⟨b,hbS,mem_iUnion₂_of_mem hij hbj⟩
obtain ⟨x,hxS,hxU,hxV⟩:=isPreconnected_closed_iff.mp hS U V hcU hcV
(hcover.trans hsplit.le) haU hbV
obtain ⟨k,hk,hxk⟩:=mem_iUnion₂.mp hxU
obtain ⟨l,hl,hxl⟩:=mem_iUnion₂.mp hxV
exact hl (hk.tail ⟨x,⟨hxS,hxk⟩,hxl⟩)
theorem connected_carrier {X Y ι : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[Finite ι] (A : ι → Set X) (K : ι → Set Y)
(hA : ∀ i,IsPreconnected (A i)) (hK : ∀ i,IsClosed (K i))
(S : Set Y) (hS : IsPreconnected S) (hcover : S⊆⋃i,K i)
(hmeet : ∀ i j,(S∩K i∩K j).Nonempty → (A i∩A j).Nonempty) :
IsPreconnected (⋃ (i : {i : ι // (S∩K i).Nonempty}),A i.val) := by
classical
apply IsPreconnected.iUnion_of_reflTransGen (s:=fun i : {i : ι // (S∩K i).Nonempty}=>A i.val) (fun i=>hA i.val)
intro i j
have hreach := closed_cover_reflTransGen (fun i : {i : ι // (S∩K i).Nonempty}=>K i.val)
(fun i=>hK i.val) S hS (by
intro y hy
obtain ⟨k,hk⟩:=mem_iUnion.mp (hcover hy)
exact mem_iUnion.mpr ⟨⟨k,⟨y,hy,hk⟩⟩,hk⟩) i j i.property j.property
exact ReflTransGen.mono (r:=fun a b : {i : ι // (S∩K i).Nonempty}=>(S∩K a.val∩K b.val).Nonempty)
(fun a b hab=>hmeet a.val b.val hab) i j hreach
lemma preimage_subset_carrier {X Y ι : Type*} (A : ι → Set X) (K : ι → Set Y)
(f : X → Y) (hcover : ∀ x,∃ i,x∈A i) (hf : ∀ i,MapsTo f (A i) (K i)) (S : Set Y) :
f ⁻¹' S⊆⋃(i : {i : ι // (S∩K i).Nonempty}),A i.val := by
intro x hx
obtain ⟨i,hi⟩:=hcover x
exact mem_iUnion.mpr ⟨⟨i,⟨f x,hx,hf i hi⟩⟩,hi⟩
lemma path_in_open_enlargement {X : Type*} [TopologicalSpace X]
[LocallyPathConnectedSpace X] {A U : Set X} (hA : IsPreconnected A)
(hU : IsOpen U) (hAU : A⊆U) {x y : X} (hx : x∈A) (hy : y∈A) :
∃ γ : Path x y,∀ t,γ t∈U := by
have hsub := hA.subset_connectedComponentIn hx hAU
have hc : IsConnected (connectedComponentIn U x) :=
isConnected_connectedComponentIn_iff.mpr (hAU hx)
have hp := hU.connectedComponentIn.isConnected_iff_isPathConnected.mp hc
have hj := hp.joinedIn x (hsub hx) y (hsub hy)
exact ⟨hj.somePath,fun t=>connectedComponentIn_subset U x (hj.somePath_mem t)⟩
end CannonFiniteCarrier
end
end
end
end
end OAI
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