feat: port SetTheory.Ordinal.Topology (#2342)

Mathlib.lean

@@ -970,6 +970,7 @@ import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.SetTheory.Lists
 import Mathlib.SetTheory.Ordinal.Arithmetic
 import Mathlib.SetTheory.Ordinal.Basic
+import Mathlib.SetTheory.Ordinal.Topology
 import Mathlib.Tactic.Abel
 import Mathlib.Tactic.Alias
 import Mathlib.Tactic.ApplyFun

Mathlib/SetTheory/Ordinal/Topology.lean

@@ -0,0 +1,236 @@
+/-
+Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+
+! This file was ported from Lean 3 source module set_theory.ordinal.topology
+! leanprover-community/mathlib commit 740acc0e6f9adf4423f92a485d0456fc271482da
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.SetTheory.Ordinal.Arithmetic
+import Mathlib.Topology.Order.Basic
+
+/-!
+### Topology of ordinals
+
+We prove some miscellaneous results involving the order topology of ordinals.
+
+### Main results
+
+* `Ordinal.isClosed_iff_sup` / `Ordinal.isClosed_iff_bsup`: A set of ordinals is closed iff it's
+  closed under suprema.
+* `Ordinal.isNormal_iff_strictMono_and_continuous`: A characterization of normal ordinal
+  functions.
+* `Ordinal.enumOrd_isNormal_iff_isClosed`: The function enumerating the ordinals of a set is
+  normal iff the set is closed.
+-/
+
+
+noncomputable section
+
+universe u v
+
+open Cardinal Order Topology
+
+namespace Ordinal
+
+variable {s : Set Ordinal.{u}} {a : Ordinal.{u}}
+
+instance : TopologicalSpace Ordinal.{u} := Preorder.topology Ordinal.{u}
+instance : OrderTopology Ordinal.{u} := ⟨rfl⟩
+
+theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) ↔ ¬IsLimit a := by
+  refine' ⟨fun h ⟨h₀, hsucc⟩ => _, fun ha => _⟩
+  · obtain ⟨b, c, hbc, hbc'⟩ :=
+      (mem_nhds_iff_exists_Ioo_subset' ⟨0, Ordinal.pos_iff_ne_zero.2 h₀⟩ ⟨_, lt_succ a⟩).1
+        (h.mem_nhds rfl)
+    have hba := hsucc b hbc.1
+    exact hba.ne (hbc' ⟨lt_succ b, hba.trans hbc.2⟩)
+  · rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha')
+    · rw [← bot_eq_zero, ← Set.Iic_bot, ← Iio_succ]
+      exact isOpen_Iio
+    · rw [← Set.Icc_self, Icc_succ_left, ← Ioo_succ_right]
+      exact isOpen_Ioo
+    · exact (ha ha').elim
+#align ordinal.is_open_singleton_iff Ordinal.isOpen_singleton_iff
+
+-- porting note: todo: generalize to a `SuccOrder`
+theorem nhds_right' (a : Ordinal) : 𝓝[>] a = ⊥ := (covby_succ a).nhdsWithin_Ioi
+
+-- todo: generalize to a `SuccOrder`
+theorem nhds_left'_eq_nhds_ne (a : Ordinal) : 𝓝[<] a = 𝓝[≠] a := by
+  rw [← nhds_left'_sup_nhds_right', nhds_right', sup_bot_eq]
+
+-- todo: generalize to a `SuccOrder`
+theorem nhds_left_eq_nhds (a : Ordinal) : 𝓝[≤] a = 𝓝 a := by
+  rw [← nhds_left_sup_nhds_right', nhds_right', sup_bot_eq]
+
+-- todo: generalize to a `SuccOrder`
+theorem nhdsBasis_Ioc (h : a ≠ 0) : (𝓝 a).HasBasis (· < a) (Set.Ioc · a) :=
+  nhds_left_eq_nhds a ▸ nhdsWithin_Iic_basis' ⟨0, h.bot_lt⟩
+
+-- todo: generalize to a `SuccOrder`
+theorem nhds_eq_pure : 𝓝 a = pure a ↔ ¬IsLimit a :=
+  (isOpen_singleton_iff_nhds_eq_pure _).symm.trans isOpen_singleton_iff
+
+-- todo: generalize `Ordinal.IsLimit` and this lemma to a `SuccOrder`
+theorem isOpen_iff : IsOpen s ↔ ∀ o ∈ s, IsLimit o → ∃ a < o, Set.Ioo a o ⊆ s := by
+  refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho => ?_
+  by_cases ho' : IsLimit o
+  · simp only [(nhdsBasis_Ioc ho'.1).mem_iff, ho', true_implies]
+    refine exists_congr fun a => and_congr_right fun ha => ?_
+    simp only [← Set.Ioo_insert_right ha, Set.insert_subset, ho, true_and]
+  · simp [nhds_eq_pure.2 ho', ho, ho']
+#align ordinal.is_open_iff Ordinal.isOpen_iff
+
+open List Set in
+theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
+    TFAE [a ∈ closure s,
+      a ∈ closure (s ∩ Iic a),
+      (s ∩ Iic a).Nonempty ∧ supₛ (s ∩ Iic a) = a,
+      ∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ supₛ t = a,
+      ∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
+        (∀ x hx, f x hx ∈ s) ∧ bsup.{u, u} o f = a,
+      ∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ sup.{u, u} f = a] := by
+  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil]
+  · simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
+    exact id
+  · intro h
+    cases' (s ∩ Iic a).eq_empty_or_nonempty with he hne
+    · simp [he] at h
+    · refine ⟨hne, (isLUB_of_mem_closure ?_ h).csupₛ_eq hne⟩
+      exact fun x hx => hx.2
+  · exact fun h => ⟨_, inter_subset_left _ _, h.1, bddAbove_Iic.mono (inter_subset_right _ _), h.2⟩
+  · rintro ⟨t, hts, hne, hbdd, rfl⟩
+    have hlub : IsLUB t (supₛ t) := isLUB_csupₛ hne hbdd
+    let ⟨y, hyt⟩ := hne
+    classical
+      refine ⟨succ (supₛ t), succ_ne_zero _, fun x _ => if x ∈ t then x else y, fun x _ => ?_, ?_⟩
+      · simp only
+        split_ifs with h <;> exact hts ‹_›
+      · refine le_antisymm (bsup_le fun x _ => ?_) (csupₛ_le hne fun x hx => ?_)
+        · split_ifs <;> exact hlub.1 ‹_›
+        · refine (if_pos hx).symm.trans_le (le_bsup _ _ <| (hlub.1 hx).trans_lt (lt_succ _))
+  · rintro ⟨o, h₀, f, hfs, rfl⟩
+    exact ⟨_, out_nonempty_iff_ne_zero.2 h₀, familyOfBFamily o f, fun _ => hfs _ _, rfl⟩
+  · rintro ⟨ι, hne, f, hfs, rfl⟩
+    rw [sup, supᵢ]
+    exact closure_mono (range_subset_iff.2 hfs) <| csupₛ_mem_closure (range_nonempty f)
+      (bddAbove_range.{u, u} f)
+
+theorem mem_closure_iff_sup :
+    a ∈ closure s ↔
+      ∃ (ι : Type u) (_ : Nonempty ι) (f : ι → Ordinal), (∀ i, f i ∈ s) ∧ sup.{u, u} f = a :=
+  ((mem_closure_tfae a s).out 0 5).trans <| by simp only [exists_prop]
+#align ordinal.mem_closure_iff_sup Ordinal.mem_closure_iff_sup
+
+theorem mem_closed_iff_sup (hs : IsClosed s) :
+    a ∈ s ↔ ∃ (ι : Type u) (_hι : Nonempty ι) (f : ι → Ordinal),
+      (∀ i, f i ∈ s) ∧ sup.{u, u} f = a :=
+  by rw [← mem_closure_iff_sup, hs.closure_eq]
+#align ordinal.mem_closed_iff_sup Ordinal.mem_closed_iff_sup
+
+theorem mem_closure_iff_bsup :
+    a ∈ closure s ↔
+      ∃ (o : Ordinal) (_ho : o ≠ 0) (f : ∀ a < o, Ordinal),
+        (∀ i hi, f i hi ∈ s) ∧ bsup.{u, u} o f = a :=
+  ((mem_closure_tfae a s).out 0 4).trans <| by simp only [exists_prop]
+#align ordinal.mem_closure_iff_bsup Ordinal.mem_closure_iff_bsup
+
+theorem mem_closed_iff_bsup (hs : IsClosed s) :
+    a ∈ s ↔
+      ∃ (o : Ordinal) (_ho : o ≠ 0) (f : ∀ a < o, Ordinal),
+        (∀ i hi, f i hi ∈ s) ∧ bsup.{u, u} o f = a :=
+  by rw [← mem_closure_iff_bsup, hs.closure_eq]
+#align ordinal.mem_closed_iff_bsup Ordinal.mem_closed_iff_bsup
+
+theorem isClosed_iff_sup :
+    IsClosed s ↔
+      ∀ {ι : Type u}, Nonempty ι → ∀ f : ι → Ordinal, (∀ i, f i ∈ s) → sup.{u, u} f ∈ s := by
+  use fun hs ι hι f hf => (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩
+  rw [← closure_subset_iff_isClosed]
+  intro h x hx
+  rcases mem_closure_iff_sup.1 hx with ⟨ι, hι, f, hf, rfl⟩
+  exact h hι f hf
+#align ordinal.is_closed_iff_sup Ordinal.isClosed_iff_sup
+
+theorem isClosed_iff_bsup :
+    IsClosed s ↔
+      ∀ {o : Ordinal}, o ≠ 0 → ∀ f : ∀ a < o, Ordinal,
+        (∀ i hi, f i hi ∈ s) → bsup.{u, u} o f ∈ s := by
+  rw [isClosed_iff_sup]
+  refine' ⟨fun H o ho f hf => H (out_nonempty_iff_ne_zero.2 ho) _ _, fun H ι hι f hf => _⟩
+  · exact fun i => hf _ _
+  · rw [← bsup_eq_sup]
+    apply H (type_ne_zero_iff_nonempty.2 hι)
+    exact fun i hi => hf _
+#align ordinal.is_closed_iff_bsup Ordinal.isClosed_iff_bsup
+
+theorem isLimit_of_mem_frontier (ha : a ∈ frontier s) : IsLimit a := by
+  simp only [frontier_eq_closure_inter_closure, Set.mem_inter_iff, mem_closure_iff] at ha
+  by_contra h
+  rw [← isOpen_singleton_iff] at h
+  rcases ha.1 _ h rfl with ⟨b, hb, hb'⟩
+  rcases ha.2 _ h rfl with ⟨c, hc, hc'⟩
+  rw [Set.mem_singleton_iff] at *
+  subst hb; subst hc
+  exact hc' hb'
+#align ordinal.is_limit_of_mem_frontier Ordinal.isLimit_of_mem_frontier
+
+theorem isNormal_iff_strictMono_and_continuous (f : Ordinal.{u} → Ordinal.{u}) :
+    IsNormal f ↔ StrictMono f ∧ Continuous f := by
+  refine' ⟨fun h => ⟨h.strictMono, _⟩, _⟩
+  · rw [continuous_def]
+    intro s hs
+    rw [isOpen_iff] at *
+    intro o ho ho'
+    rcases hs _ ho (h.isLimit ho') with ⟨a, ha, has⟩
+    rw [← IsNormal.bsup_eq.{u, u} h ho', lt_bsup] at ha
+    rcases ha with ⟨b, hb, hab⟩
+    exact
+      ⟨b, hb, fun c hc =>
+        Set.mem_preimage.2 (has ⟨hab.trans (h.strictMono hc.1), h.strictMono hc.2⟩)⟩
+  · rw [isNormal_iff_strictMono_limit]
+    rintro ⟨h, h'⟩
+    refine' ⟨h, fun o ho a h => _⟩
+    suffices : o ∈ f ⁻¹' Set.Iic a
+    exact Set.mem_preimage.1 this
+    rw [mem_closed_iff_sup (IsClosed.preimage h' (@isClosed_Iic _ _ _ _ a))]
+    exact
+      ⟨_, out_nonempty_iff_ne_zero.2 ho.1, typein (· < ·), fun i => h _ (typein_lt_self i),
+        sup_typein_limit ho.2⟩
+#align ordinal.is_normal_iff_strict_mono_and_continuous Ordinal.isNormal_iff_strictMono_and_continuous
+
+theorem enumOrd_isNormal_iff_isClosed (hs : s.Unbounded (· < ·)) :
+    IsNormal (enumOrd s) ↔ IsClosed s := by
+  have Hs := enumOrd_strictMono hs
+  refine'
+    ⟨fun h => isClosed_iff_sup.2 fun {ι} hι f hf => _, fun h =>
+      (isNormal_iff_strictMono_limit _).2 ⟨Hs, fun a ha o H => _⟩⟩
+  · let g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso hs).symm ⟨_, hf i⟩
+    suffices enumOrd s (sup.{u, u} g) = sup.{u, u} f
+      by
+      rw [← this]
+      exact enumOrd_mem hs _
+    rw [@IsNormal.sup.{u, u, u} _ h ι g hι]
+    congr
+    ext x
+    change ((enumOrdOrderIso hs) _).val = f x
+    rw [OrderIso.apply_symm_apply]
+  · rw [isClosed_iff_bsup] at h
+    suffices : enumOrd s a ≤ bsup.{u, u} a fun b (_ : b < a) => enumOrd s b
+    exact this.trans (bsup_le H)
+    cases' enumOrd_surjective hs _
+        (h ha.1 (fun b _ => enumOrd s b) fun b _ => enumOrd_mem hs b) with
+      b hb
+    rw [← hb]
+    apply Hs.monotone
+    by_contra' hba
+    apply (Hs (lt_succ b)).not_le
+    rw [hb]
+    exact le_bsup.{u, u} _ _ (ha.2 _ hba)
+#align ordinal.enum_ord_is_normal_iff_is_closed Ordinal.enumOrd_isNormal_iff_isClosed
+
+end Ordinal
+