| import Mathlib.Topology.MetricSpace.Polish |
| import Mathlib.Topology.GDelta.MetrizableSpace |
| import Mathlib.Topology.Sequences |
|
|
| namespace OAI |
|
|
| /-! A completely metrizable subspace of a metrizable space is a G-delta. |
| The open sets below bound the diameter of their inverse images. A sequence |
| approaching a point in all these sets is Cauchy in the complete source metric. -/ |
| open Set Function Filter Topology TopologicalSpace |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma isGDelta_range_embedding |
| {A B : Type*} [TopologicalSpace A] [IsCompletelyMetrizableSpace A] |
| [TopologicalSpace B] [MetrizableSpace B] {f : A → B} (hf : IsEmbedding f) : |
| IsGδ (range f) := by |
| let := upgradeIsCompletelyMetrizable A |
| let : MetricSpace B := metrizableSpaceMetric B |
| let U : ℕ → Set B := fun n => ⋃₀ {V : Set B | IsOpen V ∧ |
| ∀ a b : A, f a ∈ V → f b ∈ V → dist a b < 1 / (n + 1 : ℝ)} |
| have hU : ∀ n, IsOpen (U n) := fun n => isOpen_sUnion fun V hV => hV.1 |
| have hsub : ∀ n, range f ⊆ U n := by |
| intro n y hy |
| obtain ⟨x, rfl⟩ := hy |
| obtain ⟨V, hVo, hV⟩ := hf.isInducing.isOpen_iff.mp |
| (Metric.isOpen_ball (x := x) (ε := (1 / (n + 1 : ℝ)) / 2)) |
| refine mem_sUnion.mpr ⟨V, ⟨hVo, ?_⟩, ?_⟩ |
| · intro a b ha hb |
| have ha' : dist a x < (1 / (n + 1 : ℝ)) / 2 := by |
| have haa : a ∈ f ⁻¹' V := ha |
| rw [hV] at haa |
| exact haa |
| have hb' : dist b x < (1 / (n + 1 : ℝ)) / 2 := by |
| have hbb : b ∈ f ⁻¹' V := hb |
| rw [hV] at hbb |
| exact hbb |
| linarith [dist_triangle_right a b x] |
| · have hx : x ∈ Metric.ball x ((1 / (n + 1 : ℝ)) / 2) := |
| Metric.mem_ball_self (by positivity) |
| rwa [← hV] at hx |
| have heq : range f = closure (range f) ∩ ⋂ n, U n := by |
| apply Subset.antisymm |
| · exact fun y hy => ⟨subset_closure hy, mem_iInter.mpr (fun n => hsub n hy)⟩ |
| · intro y hy |
| obtain ⟨ys, hys, hlim⟩ := mem_closure_iff_seq_limit.mp hy.1 |
| choose xs hxs using hys |
| have hflim : Tendsto (fun n => f (xs n)) atTop (𝓝 y) := by |
| simpa only [hxs] using hlim |
| have hc : CauchySeq xs := Metric.cauchySeq_iff.mpr (by |
| intro ε hε |
| obtain ⟨n, hn⟩ := exists_nat_one_div_lt hε |
| obtain ⟨V, hV, hyV⟩ := mem_sUnion.mp (mem_iInter.mp hy.2 n) |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (hflim (hV.1.mem_nhds hyV)) |
| exact ⟨N, fun i hi j hj => (hV.2 _ _ (hN i hi) (hN j hj)).trans hn⟩) |
| obtain ⟨x, hx⟩ := cauchySeq_tendsto_of_complete hc |
| exact ⟨x, tendsto_nhds_unique (hf.continuous.tendsto x |>.comp hx) hflim⟩ |
| rw [heq] |
| exact isClosed_closure.isGδ.inter (IsGδ.iInter_of_isOpen hU) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
