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9655094 | 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 | 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
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