| import OAI.Analysis.CircleDomains.Topology.ComponentTopology |
| import Mathlib.Topology.Separation.Regular |
|
|
| namespace OAI |
|
|
| /-! |
| # Closedness of the component-collapse relation |
|
|
| The connected-component space of a closed set in a compact Hausdorff space |
| is compact Hausdorff. Collapsing its components gives a closed relation, |
| as required by Moore's sphere decomposition theorem. |
| -/ |
|
|
| open Set Topology |
|
|
| namespace CircleDomainRigidity |
|
|
| /-- Collapsing the connected components of a closed set in a compact Hausdorff |
| space, and leaving the remaining points fixed, gives a closed relation. -/ |
| theorem isClosed_componentCollapseRelation |
| {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X] |
| {F : Set X} (hF : IsClosed F) : |
| IsClosed {p : X × X | p.1 = p.2 ∨ |
| (p.1 ∈ F ∧ p.2 ∈ connectedComponentIn F p.1)} := by |
| let : CompactSpace F := isCompact_iff_compactSpace.mp hF.isCompact |
| let K : Set (F × F) := {p | |
| ConnectedComponents.mk p.2 = ConnectedComponents.mk p.1} |
| have hK : IsClosed K := |
| isClosed_eq (ConnectedComponents.continuous_coe.comp continuous_snd) |
| (ConnectedComponents.continuous_coe.comp continuous_fst) |
| let valPair : F × F → X × X := fun p => (p.1.val, p.2.val) |
| have hval : Continuous valPair := |
| (continuous_subtype_val.comp continuous_fst).prodMk |
| (continuous_subtype_val.comp continuous_snd) |
| have himage : IsClosed (valPair '' K) := (hK.isCompact.image hval).isClosed |
| have hdiag : IsClosed {p : X × X | p.1 = p.2} := |
| isClosed_eq continuous_fst continuous_snd |
| convert hdiag.union himage using 1 |
| ext p |
| constructor |
| · rintro (heq | ⟨hx, hy⟩) |
| · exact Or.inl heq |
| · right |
| rw [connectedComponentIn_eq_image hx] at hy |
| obtain ⟨y, hy, hvaly⟩ := hy |
| refine ⟨(⟨p.1, hx⟩, y), ?_, ?_⟩ |
| · exact ConnectedComponents.coe_eq_coe'.mpr hy |
| · exact Prod.ext rfl hvaly |
| · rintro (heq | ⟨⟨x, y⟩, hxy, hpair⟩) |
| · exact Or.inl heq |
| · have hx : x.val = p.1 := congrArg Prod.fst hpair |
| have hy : y.val = p.2 := congrArg Prod.snd hpair |
| right |
| refine ⟨hx ▸ x.property, ?_⟩ |
| have hmem : y ∈ connectedComponent x := ConnectedComponents.coe_eq_coe'.mp hxy |
| have hin : y.val ∈ connectedComponentIn F x.val := by |
| rw [connectedComponentIn_eq_image x.property] |
| exact ⟨y, hmem, rfl⟩ |
| simpa only [hx, hy] using hin |
|
|
| end CircleDomainRigidity |
|
|
| end OAI |
|
|
