| import Mathlib.Topology.ExtendFrom |
|
|
| namespace OAI |
|
|
| /-! Local continuous extensions provide a single canonical extension |
| to the closure. This glues boundary values without choosing compatible |
| values by hand. -/ |
| open Set Function Filter |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma tendsto_of_local_extension {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| {A O : Set X} {f g : X → Y} (hO : IsOpen O) {x : X} (hx : x∈O) |
| (hg : ContinuousOn g O) (he : EqOn g f (O∩A)) : |
| Tendsto f (𝓝[A] x) (𝓝 (g x)) := by |
| apply ((hg.continuousAt (hO.mem_nhds hx)).tendsto.mono_left nhdsWithin_le_nhds).congr' |
| filter_upwards [mem_nhdsWithin_of_mem_nhds (hO.mem_nhds hx),self_mem_nhdsWithin] with z hzO hzA |
| exact he ⟨hzO,hzA⟩ |
|
|
| lemma continuousOn_extendFrom_of_local_extensions {X Y : Type*} |
| [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] |
| {A K : Set X} {f : X → Y} (hK : K⊆closure A) |
| (hl : ∀x∈K,∃(O:Set X)(g:X→Y),IsOpen O ∧ x∈O ∧ ContinuousOn g O ∧ EqOn g f (O∩A)) : |
| ContinuousOn (extendFrom A f) K := by |
| apply continuousOn_extendFrom hK |
| intro x hx |
| obtain ⟨O,g,hO,hxO,hg,he⟩ := hl x hx |
| exact ⟨g x,tendsto_of_local_extension hO hxO hg he⟩ |
|
|
| lemma extendFrom_eq_local_extension {X Y : Type*} |
| [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] |
| {A O : Set X} {f g : X → Y} (hO : IsOpen O) (hg : ContinuousOn g O) |
| (he : EqOn g f (O∩A)) : EqOn (extendFrom A f) g (O∩closure A) := by |
| intro x hx |
| exact extendFrom_eq hx.2 (tendsto_of_local_extension hO hx.1 hg he) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
