sorgenfrey_line ⟷ Counterexamples.SorgenfreyLine

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -8,7 +8,7 @@ import Topology.Algebra.Order.Archimedean
 import Topology.Compactness.Paracompact
 import Topology.Metrizable.Basic
 import Topology.EMetricSpace.Paracompact
-import Data.Set.Intervals.Monotone
+import Order.Interval.Set.Monotone
 
 #align_import sorgenfrey_line from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -5,9 +5,9 @@ Authors: Yury Kudryashov
 -/
 import Topology.Instances.Irrational
 import Topology.Algebra.Order.Archimedean
-import Topology.Paracompact
-import Topology.MetricSpace.Metrizable
-import Topology.MetricSpace.EmetricParacompact
+import Topology.Compactness.Paracompact
+import Topology.Metrizable.Basic
+import Topology.EMetricSpace.Paracompact
 import Data.Set.Intervals.Monotone
 
 #align_import sorgenfrey_line from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
@@ -127,7 +127,7 @@ theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x
   isOpen_iff_mem_nhds.trans <| forall₂_congr fun x hx => (nhds_basis_Ico x).mem_iff
 #align counterexample.sorgenfrey_line.is_open_iff Counterexample.SorgenfreyLine.isOpen_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
 theorem isClosed_iff {s : Set ℝₗ} : IsClosed s ↔ ∀ (x) (_ : x ∉ s), ∃ y > x, Disjoint (Ico x y) s :=
   by simp only [← isOpen_compl_iff, is_open_iff, mem_compl_iff, subset_compl_iff_disjoint_right]
 #align counterexample.sorgenfrey_line.is_closed_iff Counterexample.SorgenfreyLine.isClosed_iff

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -105,8 +105,8 @@ theorem nhds_basis_Ico_inv_pNat (a : ℝₗ) :
     (nhds_basis_Ico a).to_hasBasis (fun b hb => _) fun n hn =>
       ⟨_, lt_add_of_pos_right _ (inv_pos.2 <| Nat.cast_pos.2 n.Pos), subset.rfl⟩
   rcases exists_nat_one_div_lt (sub_pos.2 hb) with ⟨k, hk⟩
-  rw [one_div] at hk 
-  rw [← Nat.cast_add_one] at hk 
+  rw [one_div] at hk
+  rw [← Nat.cast_add_one] at hk
   exact ⟨k.succ_pnat, trivial, Ico_subset_Ico_right (le_sub_iff_add_le'.1 hk.le)⟩
 #align counterexample.sorgenfrey_line.nhds_basis_Ico_inv_pnat Counterexample.SorgenfreyLine.nhds_basis_Ico_inv_pNat
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -183,8 +183,8 @@ theorem isClopen_Ico (a b : ℝₗ) : IsClopen (Ico a b) :=
 
 instance : TotallyDisconnectedSpace ℝₗ :=
   ⟨fun s hs' hs x hx y hy =>
-    le_antisymm (hs.subset_clopen (isClopen_Ici x) ⟨x, hx, le_rfl⟩ hy)
-      (hs.subset_clopen (isClopen_Ici y) ⟨y, hy, le_rfl⟩ hx)⟩
+    le_antisymm (hs.subset_isClopen (isClopen_Ici x) ⟨x, hx, le_rfl⟩ hy)
+      (hs.subset_isClopen (isClopen_Ici y) ⟨y, hy, le_rfl⟩ hx)⟩
 
 instance : FirstCountableTopology ℝₗ :=
   ⟨fun x => (nhds_basis_Ico_rat x).IsCountablyGenerated⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe

@@ -85,7 +85,7 @@ theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (fun b => a < b) fun b =
     iInf_inf, this, iInf_subtype]
   suffices : (⨅ x ∈ Ioi a, 𝓟 (Iio x)).HasBasis ((· < ·) a) Iio; exact this.principal_inf _
   refine' has_basis_binfi_principal _ nonempty_Ioi
-  exact directedOn_iff_directed.2 (directed_of_inf fun x y hxy => Iio_subset_Iio hxy)
+  exact directedOn_iff_directed.2 (directed_of_isDirected_ge fun x y hxy => Iio_subset_Iio hxy)
 #align counterexample.sorgenfrey_line.nhds_basis_Ico Counterexample.SorgenfreyLine.nhds_basis_Ico
 
 theorem nhds_basis_Ico_rat (a : ℝₗ) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,12 +3,12 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Topology.Instances.Irrational
-import Mathbin.Topology.Algebra.Order.Archimedean
-import Mathbin.Topology.Paracompact
-import Mathbin.Topology.MetricSpace.Metrizable
-import Mathbin.Topology.MetricSpace.EmetricParacompact
-import Mathbin.Data.Set.Intervals.Monotone
+import Topology.Instances.Irrational
+import Topology.Algebra.Order.Archimedean
+import Topology.Paracompact
+import Topology.MetricSpace.Metrizable
+import Topology.MetricSpace.EmetricParacompact
+import Data.Set.Intervals.Monotone
 
 #align_import sorgenfrey_line from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 
@@ -127,7 +127,7 @@ theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x
   isOpen_iff_mem_nhds.trans <| forall₂_congr fun x hx => (nhds_basis_Ico x).mem_iff
 #align counterexample.sorgenfrey_line.is_open_iff Counterexample.SorgenfreyLine.isOpen_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
 theorem isClosed_iff {s : Set ℝₗ} : IsClosed s ↔ ∀ (x) (_ : x ∉ s), ∃ y > x, Disjoint (Ico x y) s :=
   by simp only [← isOpen_compl_iff, is_open_iff, mem_compl_iff, subset_compl_iff_disjoint_right]
 #align counterexample.sorgenfrey_line.is_closed_iff Counterexample.SorgenfreyLine.isClosed_iff

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module sorgenfrey_line
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Instances.Irrational
 import Mathbin.Topology.Algebra.Order.Archimedean
@@ -15,6 +10,8 @@ import Mathbin.Topology.MetricSpace.Metrizable
 import Mathbin.Topology.MetricSpace.EmetricParacompact
 import Mathbin.Data.Set.Intervals.Monotone
 
+#align_import sorgenfrey_line from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # Sorgenfrey line
 
@@ -130,7 +127,7 @@ theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x
   isOpen_iff_mem_nhds.trans <| forall₂_congr fun x hx => (nhds_basis_Ico x).mem_iff
 #align counterexample.sorgenfrey_line.is_open_iff Counterexample.SorgenfreyLine.isOpen_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
 theorem isClosed_iff {s : Set ℝₗ} : IsClosed s ↔ ∀ (x) (_ : x ∉ s), ∃ y > x, Disjoint (Ico x y) s :=
   by simp only [← isOpen_compl_iff, is_open_iff, mem_compl_iff, subset_compl_iff_disjoint_right]
 #align counterexample.sorgenfrey_line.is_closed_iff Counterexample.SorgenfreyLine.isClosed_iff

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module sorgenfrey_line
-! leanprover-community/mathlib commit 328375597f2c0dd00522d9c2e5a33b6a6128feeb
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Data.Set.Intervals.Monotone
 /-!
 # Sorgenfrey line
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define `sorgenfrey_line` (notation: `ℝₗ`) to be the Sorgenfrey line. It is the real
 line with the topology space structure generated by half-open intervals `set.Ico a b`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -51,7 +51,6 @@ def SorgenfreyLine : Type :=
 deriving ConditionallyCompleteLinearOrder, LinearOrderedField, Archimedean
 #align counterexample.sorgenfrey_line Counterexample.SorgenfreyLine
 
--- mathport name: sorgenfrey_line
 scoped[SorgenfreyLine] notation "ℝₗ" => SorgenfreyLine
 
 namespace SorgenfreyLine

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -214,13 +214,11 @@ instance : T5Space ℝₗ :=
       min (X x) (Y y) ≤ X x := min_le_left _ _
       _ ≤ y := (not_lt.1 fun hyx => (hXd x hx).le_bot ⟨⟨hle, hyx⟩, subset_closure hy⟩)
       _ ≤ max x y := le_max_right _ _
-      
   ·
     calc
       min (X x) (Y y) ≤ Y y := min_le_right _ _
       _ ≤ x := (not_lt.1 fun hxy => (hYd y hy).le_bot ⟨⟨hle, hxy⟩, subset_closure hx⟩)
       _ ≤ max x y := le_max_left _ _
-      
 
 theorem denseRange_coe_rat : DenseRange (coe : ℚ → ℝₗ) :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -128,7 +128,7 @@ theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x
   isOpen_iff_mem_nhds.trans <| forall₂_congr fun x hx => (nhds_basis_Ico x).mem_iff
 #align counterexample.sorgenfrey_line.is_open_iff Counterexample.SorgenfreyLine.isOpen_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
 theorem isClosed_iff {s : Set ℝₗ} : IsClosed s ↔ ∀ (x) (_ : x ∉ s), ∃ y > x, Disjoint (Ico x y) s :=
   by simp only [← isOpen_compl_iff, is_open_iff, mem_compl_iff, subset_compl_iff_disjoint_right]
 #align counterexample.sorgenfrey_line.is_closed_iff Counterexample.SorgenfreyLine.isClosed_iff

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

Move Set.Ixx, Finset.Ixx, Multiset.Ixx together under two different folders:

• Order.Interval for their definition and basic properties
• Algebra.Order.Interval for their algebraic properties

Move the definitions of Multiset.Ixx to what is now Order.Interval.Multiset. I believe we could just delete this file in a later PR as nothing uses it (and I already had doubts when defining Multiset.Ixx three years ago).

Move the algebraic results out of what is now Order.Interval.Finset.Basic to a new file Algebra.Order.Interval.Finset.Basic.

@@ -3,12 +3,12 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
+import Mathlib.Order.Interval.Set.Monotone
 import Mathlib.Topology.Instances.Irrational
 import Mathlib.Topology.Algebra.Order.Archimedean
 import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.Metrizable.Urysohn
 import Mathlib.Topology.EMetricSpace.Paracompact
-import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal
 import Mathlib.Topology.Baire.Lemmas
 import Mathlib.Topology.Baire.LocallyCompactRegular

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -217,16 +217,16 @@ instance : T5Space ℝₗ := by
       _ ≤ x := (not_lt.1 fun hxy => (hYd y hy).le_bot ⟨⟨hle, hxy⟩, subset_closure hx⟩)
       _ ≤ max x y := le_max_left _ _
 
-theorem denseRange_coe_rat : DenseRange ((↑) : ℚ → ℝₗ) := by
+theorem denseRange_ratCast : DenseRange ((↑) : ℚ → ℝₗ) := by
   refine' dense_iff_inter_open.2 _
   rintro U Uo ⟨x, hx⟩
   rcases isOpen_iff.1 Uo _ hx with ⟨y, hxy, hU⟩
   rcases exists_rat_btwn hxy with ⟨z, hxz, hzy⟩
   exact ⟨z, hU ⟨hxz.le, hzy⟩, mem_range_self _⟩
-#align counterexample.sorgenfrey_line.dense_range_coe_rat Counterexample.SorgenfreyLine.denseRange_coe_rat
+#align counterexample.sorgenfrey_line.dense_range_coe_rat Counterexample.SorgenfreyLine.denseRange_ratCast
 
 instance : SeparableSpace ℝₗ :=
-  ⟨⟨_, countable_range _, denseRange_coe_rat⟩⟩
+  ⟨⟨_, countable_range _, denseRange_ratCast⟩⟩
 
 theorem isClosed_antidiagonal (c : ℝₗ) : IsClosed {x : ℝₗ × ℝₗ | x.1 + x.2 = c} :=
   isClosed_singleton.preimage continuous_add

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -84,7 +84,7 @@ theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (a < ·) (Ico a ·) := b
   haveI : Nonempty { x // x ≤ a } := Set.nonempty_Iic_subtype
   have : (⨅ x : { i // i ≤ a }, 𝓟 (Ici ↑x)) = 𝓟 (Ici a) := by
     refine' (IsLeast.isGLB _).iInf_eq
-    exact ⟨⟨⟨a, le_rfl⟩, rfl⟩, forall_range_iff.2 fun b => principal_mono.2 <| Ici_subset_Ici.2 b.2⟩
+    exact ⟨⟨⟨a, le_rfl⟩, rfl⟩, forall_mem_range.2 fun b => principal_mono.2 <| Ici_subset_Ici.2 b.2⟩
   simp only [mem_setOf_eq, iInf_and, iInf_exists, @iInf_comm _ (_ ∈ _), @iInf_comm _ (Set ℝₗ),
     iInf_iInf_eq_right, mem_Ico]
   simp_rw [@iInf_comm _ ℝₗ (_ ≤ _), iInf_subtype', ← Ici_inter_Iio, ← inf_principal,

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -47,7 +47,7 @@ noncomputable section
 /-- The Sorgenfrey line. It is the real line with the topology space structure generated by
 half-open intervals `Set.Ico a b`. -/
 def SorgenfreyLine : Type := ℝ
--- porting note: was deriving ConditionallyCompleteLinearOrder, LinearOrderedField, Archimedean
+-- Porting note: was deriving ConditionallyCompleteLinearOrder, LinearOrderedField, Archimedean
 #align counterexample.sorgenfrey_line Counterexample.SorgenfreyLine
 
 @[inherit_doc]

• move definition to Topology/Defs/Basic;
• move lemmas to Topology/Baire/Lemmas;
• move instances to Topology/Baire/CompleteMetrizable and Topology/Baire/LocallyCompactRegular;
• assume [UniformSpace X] [IsCountablyGenerated (𝓤 X)] instead of [PseudoMetricSpace X] in the 1st theorem.

This way Lemmas file does not depend on analysis.

@@ -10,7 +10,8 @@ import Mathlib.Topology.Metrizable.Urysohn
 import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal
-import Mathlib.Topology.MetricSpace.Baire
+import Mathlib.Topology.Baire.Lemmas
+import Mathlib.Topology.Baire.LocallyCompactRegular
 
 #align_import sorgenfrey_line from "leanprover-community/mathlib"@"328375597f2c0dd00522d9c2e5a33b6a6128feeb"
 

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -315,14 +315,14 @@ theorem not_separatedNhds_rat_irrational_antidiag :
   have Hd : Dense (⋃ n, interior (C n)) :=
     IsGδ.setOf_irrational.dense_iUnion_interior_of_closed dense_irrational
       (fun _ => isClosed_closure) H
-  obtain ⟨N, hN⟩ : ∃ n : ℕ+, (interior <| C n).Nonempty; exact nonempty_iUnion.mp Hd.nonempty
+  obtain ⟨N, hN⟩ : ∃ n : ℕ+, (interior <| C n).Nonempty := nonempty_iUnion.mp Hd.nonempty
   /- Choose a rational number `r` in the interior of the closure of `C N`, then choose `n ≥ N > 0`
     such that `Ico r (r + n⁻¹) × Ico (-r) (-r + n⁻¹) ⊆ U`. -/
   rcases Rat.denseRange_cast.exists_mem_open isOpen_interior hN with ⟨r, hr⟩
   have hrU : ((r, -r) : ℝₗ × ℝₗ) ∈ U := @SU (r, -r) ⟨add_neg_self _, r, rfl⟩
   obtain ⟨n, hnN, hn⟩ :
-    ∃ n, N ≤ n ∧ Ico (r : ℝₗ) (r + (n : ℝₗ)⁻¹) ×ˢ Ico (-r : ℝₗ) (-r + (n : ℝₗ)⁻¹) ⊆ U
-  exact ((nhds_prod_antitone_basis_inv_pnat _ _).hasBasis_ge N).mem_iff.1 (Uo.mem_nhds hrU)
+      ∃ n, N ≤ n ∧ Ico (r : ℝₗ) (r + (n : ℝₗ)⁻¹) ×ˢ Ico (-r : ℝₗ) (-r + (n : ℝₗ)⁻¹) ⊆ U :=
+    ((nhds_prod_antitone_basis_inv_pnat _ _).hasBasis_ge N).mem_iff.1 (Uo.mem_nhds hrU)
   /- Finally, choose `x ∈ Ioo (r : ℝ) (r + n⁻¹) ∩ C N`. Then `(x, -r)` belongs both to `U` and `V`,
     so they are not disjoint. This contradiction completes the proof. -/
   obtain ⟨x, hxn, hx_irr, rfl⟩ :

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -88,8 +88,8 @@ theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (a < ·) (Ico a ·) := b
     iInf_iInf_eq_right, mem_Ico]
   simp_rw [@iInf_comm _ ℝₗ (_ ≤ _), iInf_subtype', ← Ici_inter_Iio, ← inf_principal,
     ← inf_iInf, ← iInf_inf, this, iInf_subtype]
-  suffices : (⨅ x ∈ Ioi a, 𝓟 (Iio x)).HasBasis (a < ·) Iio; exact this.principal_inf _
-  refine' hasBasis_biInf_principal _ nonempty_Ioi
+  suffices (⨅ x ∈ Ioi a, 𝓟 (Iio x)).HasBasis (a < ·) Iio from this.principal_inf _
+  refine hasBasis_biInf_principal ?_ nonempty_Ioi
   exact directedOn_iff_directed.2 <| Monotone.directed_ge fun x y hxy ↦ Iio_subset_Iio hxy
 #align counterexample.sorgenfrey_line.nhds_basis_Ico Counterexample.SorgenfreyLine.nhds_basis_Ico
 

Rename many isGδ_some lemmas to IsGδ.some. Also resolve a TODO.

@@ -10,6 +10,7 @@ import Mathlib.Topology.Metrizable.Urysohn
 import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal
+import Mathlib.Topology.MetricSpace.Baire
 
 #align_import sorgenfrey_line from "leanprover-community/mathlib"@"328375597f2c0dd00522d9c2e5a33b6a6128feeb"
 
@@ -312,7 +313,8 @@ theorem not_separatedNhds_rat_irrational_antidiag :
   have H : {x : ℝ | Irrational x} ⊆ ⋃ n, C n := fun x hx =>
     mem_iUnion.2 ⟨_, subset_closure ⟨hx, rfl⟩⟩
   have Hd : Dense (⋃ n, interior (C n)) :=
-    isGδ_irrational.dense_iUnion_interior_of_closed dense_irrational (fun _ => isClosed_closure) H
+    IsGδ.setOf_irrational.dense_iUnion_interior_of_closed dense_irrational
+      (fun _ => isClosed_closure) H
   obtain ⟨N, hN⟩ : ∃ n : ℕ+, (interior <| C n).Nonempty; exact nonempty_iUnion.mp Hd.nonempty
   /- Choose a rational number `r` in the interior of the closure of `C N`, then choose `n ≥ N > 0`
     such that `Ico r (r + n⁻¹) × Ico (-r) (-r + n⁻¹) ⊆ U`. -/

• upgrade isSeparable_iUnion to an Iff lemma, restore the original version as IsSeparable.iUnion;
• add isSeparable_union and isSeparable_closure;
• upgrade isSeparable_pi from [Finite ι] to [Countable ι], add IsSeparable.univ_pi version;
• add Dense.isSeparable_iff and isSeparable_range;
• rename isSeparable_of_separableSpace_subtype to IsSeparable.of_subtype;
• rename isSeparable_of_separableSpace to IsSeparable.of_separableSpace.

@@ -270,7 +270,7 @@ theorem not_normalSpace_prod : ¬NormalSpace (ℝₗ × ℝₗ) :=
 
 /-- An antidiagonal is a separable set but is not a separable space. -/
 theorem isSeparable_antidiagonal (c : ℝₗ) : IsSeparable {x : ℝₗ × ℝₗ | x.1 + x.2 = c} :=
-  isSeparable_of_separableSpace _
+  .of_separableSpace _
 
 /-- An antidiagonal is a separable set but is not a separable space. -/
 theorem not_separableSpace_antidiagonal (c : ℝₗ) :

Make all the notations that unambiguously should inherit the docstring of their definition actually inherit it.

Also write a few docstrings by hand. I only wrote the ones I was competent to write and which I was sure of. Some docstrings come from mathlib3 as they were lost during the early port.

This PR is only intended as a first pass There are many more docstrings to add.

@@ -48,6 +48,7 @@ def SorgenfreyLine : Type := ℝ
 -- porting note: was deriving ConditionallyCompleteLinearOrder, LinearOrderedField, Archimedean
 #align counterexample.sorgenfrey_line Counterexample.SorgenfreyLine
 
+@[inherit_doc]
 scoped[SorgenfreyLine] notation "ℝₗ" => Counterexample.SorgenfreyLine
 open scoped SorgenfreyLine
 

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -166,7 +166,7 @@ instance : ContinuousAdd ℝₗ := by
   exact (continuous_add.tendsto _).inf (MapsTo.tendsto fun x hx => add_le_add hx.1 hx.2)
 
 theorem isClopen_Ici (a : ℝₗ) : IsClopen (Ici a) :=
-  ⟨isOpen_Ici a, isClosed_Ici⟩
+  ⟨isClosed_Ici, isOpen_Ici a⟩
 #align counterexample.sorgenfrey_line.is_clopen_Ici Counterexample.SorgenfreyLine.isClopen_Ici
 
 theorem isClopen_Iio (a : ℝₗ) : IsClopen (Iio a) := by
@@ -247,7 +247,7 @@ theorem isClosed_of_subset_antidiagonal {s : Set (ℝₗ × ℝₗ)} {c : ℝₗ
   obtain rfl : x + y = c := by
     change (x, y) ∈ {p : ℝₗ × ℝₗ | p.1 + p.2 = c}
     exact closure_minimal (hs : s ⊆ {x | x.1 + x.2 = c}) (isClosed_antidiagonal c) H
-  rcases mem_closure_iff.1 H (Ici (x, y)) (isClopen_Ici_prod _).1 left_mem_Ici with
+  rcases mem_closure_iff.1 H (Ici (x, y)) (isClopen_Ici_prod _).2 left_mem_Ici with
     ⟨⟨x', y'⟩, ⟨hx : x ≤ x', hy : y ≤ y'⟩, H⟩
   convert H
   · refine' hx.antisymm _

See Zulip thread for the discussion.

@@ -118,7 +118,7 @@ theorem nhds_antitone_basis_Ico_inv_pnat (a : ℝₗ) :
     (𝓝 a).HasAntitoneBasis fun n : ℕ+ => Ico a (a + (n : ℝₗ)⁻¹) :=
   ⟨nhds_basis_Ico_inv_pnat a, monotone_const.Ico <| Antitone.const_add
     (fun k _l hkl => inv_le_inv_of_le (Nat.cast_pos.2 k.2)
-      (Nat.mono_cast $ Subtype.coe_le_coe.2 hkl)) _⟩
+      (Nat.mono_cast <| Subtype.coe_le_coe.2 hkl)) _⟩
 #align counterexample.sorgenfrey_line.nhds_antitone_basis_Ico_inv_pnat Counterexample.SorgenfreyLine.nhds_antitone_basis_Ico_inv_pnat
 
 theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x y ⊆ s :=

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -204,7 +204,7 @@ instance : T5Space ℝₗ := by
     (bUnion_mem_nhdsSet fun y hy => (isOpen_Ico y (Y y)).mem_nhds <| left_mem_Ico.2 (hY y hy))
   simp only [disjoint_iUnion_left, disjoint_iUnion_right, Ico_disjoint_Ico]
   intro y hy x hx
-  cases' le_total x y with hle hle
+  rcases le_total x y with hle | hle
   · calc
       min (X x) (Y y) ≤ X x := min_le_left _ _
       _ ≤ y := (not_lt.1 fun hyx => (hXd x hx).le_bot ⟨⟨hle, hyx⟩, subset_closure hy⟩)

This PR renames the field Clopens.clopen' -> Clopens.isClopen', and the lemmas

• preimage_closed_of_closed -> ContinuousOn.preimage_isClosed_of_isClosed

as well as: ClopenUpperSet.clopen -> ClopenUpperSet.isClopen connectedComponent_eq_iInter_clopen -> connectedComponent_eq_iInter_isClopen connectedComponent_subset_iInter_clopen -> connectedComponent_subset_iInter_isClopen continuous_boolIndicator_iff_clopen -> continuous_boolIndicator_iff_isClopen continuousOn_boolIndicator_iff_clopen -> continuousOn_boolIndicator_iff_isClopen DiscreteQuotient.ofClopen -> DiscreteQuotient.ofIsClopen disjoint_or_subset_of_clopen -> disjoint_or_subset_of_isClopen exists_clopen_{lower,upper}of_not_le -> exists_isClopen{lower,upper}_of_not_le exists_clopen_of_cofiltered -> exists_isClopen_of_cofiltered exists_clopen_of_totally_separated -> exists_isClopen_of_totally_separated exists_clopen_upper_or_lower_of_ne -> exists_isClopen_upper_or_lower_of_ne IsPreconnected.subset_clopen -> IsPreconnected.subset_isClopen isTotallyDisconnected_of_clopen_set -> isTotallyDisconnected_of_isClopen_set LocallyConstant.ofClopen_fiber_one -> LocallyConstant.ofIsClopen_fiber_one LocallyConstant.ofClopen_fiber_zero -> LocallyConstant.ofIsClopen_fiber_zero LocallyConstant.ofClopen -> LocallyConstant.ofIsClopen preimage_clopen_of_clopen -> preimage_isClopen_of_isClopen TopologicalSpace.Clopens.clopen -> TopologicalSpace.Clopens.isClopen

@@ -179,8 +179,8 @@ theorem isClopen_Ico (a b : ℝₗ) : IsClopen (Ico a b) :=
 
 instance : TotallyDisconnectedSpace ℝₗ :=
   ⟨fun _ _ hs x hx y hy =>
-    le_antisymm (hs.subset_clopen (isClopen_Ici x) ⟨x, hx, left_mem_Ici⟩ hy)
-      (hs.subset_clopen (isClopen_Ici y) ⟨y, hy, left_mem_Ici⟩ hx)⟩
+    le_antisymm (hs.subset_isClopen (isClopen_Ici x) ⟨x, hx, left_mem_Ici⟩ hy)
+      (hs.subset_isClopen (isClopen_Ici y) ⟨y, hy, left_mem_Ici⟩ hx)⟩
 
 instance : FirstCountableTopology ℝₗ :=
   ⟨fun x => (nhds_basis_Ico_rat x).isCountablyGenerated⟩

Move

• basic definitions to Topology.Metrizable.Basic,
• Urysohn's metrization theorem to `Topology.Metrizable.Urysohns', and
• metrizability of a uniform space with countably generated uniformity to Topology.Metrizable.Uniform.

The next step is to redefine Metrizable as "uniformizable with countably generated uniformity" and make this definition available much earlier.

@@ -6,7 +6,7 @@ Authors: Yury Kudryashov
 import Mathlib.Topology.Instances.Irrational
 import Mathlib.Topology.Algebra.Order.Archimedean
 import Mathlib.Topology.Compactness.Paracompact
-import Mathlib.Topology.MetricSpace.Metrizable
+import Mathlib.Topology.Metrizable.Urysohn
 import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal

New lemmas / instances

• An archimedean ordered semiring is directed upwards.
• Filter.hasAntitoneBasis_atTop;
• Filter.HasAntitoneBasis.iInf_principal;

Fix typos

• Docstrings: "if the agree" -> "if they agree".
• ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self -> ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self.

Weaken typeclass assumptions

From a semilattice to a directed type

• MeasureTheory.tendsto_measure_iUnion;
• MeasureTheory.tendsto_measure_iInter;
• Monotone.directed_le, Monotone.directed_ge;
• Antitone.directed_le, Antitone.directed_ge;
• directed_of_sup, renamed to directed_of_isDirected_le;
• directed_of_inf, renamed to directed_of_isDirected_ge;

From a strict ordered semiring to an ordered semiring

• tendsto_nat_cast_atTop_atTop;
• Filter.Eventually.nat_cast_atTop;
• atTop_hasAntitoneBasis_of_archimedean;

@@ -88,7 +88,7 @@ theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (a < ·) (Ico a ·) := b
     ← inf_iInf, ← iInf_inf, this, iInf_subtype]
   suffices : (⨅ x ∈ Ioi a, 𝓟 (Iio x)).HasBasis (a < ·) Iio; exact this.principal_inf _
   refine' hasBasis_biInf_principal _ nonempty_Ioi
-  exact directedOn_iff_directed.2 (directed_of_inf fun x y hxy => Iio_subset_Iio hxy)
+  exact directedOn_iff_directed.2 <| Monotone.directed_ge fun x y hxy ↦ Iio_subset_Iio hxy
 #align counterexample.sorgenfrey_line.nhds_basis_Ico Counterexample.SorgenfreyLine.nhds_basis_Ico
 
 theorem nhds_basis_Ico_rat (a : ℝₗ) :

Split up the 2000-line Topology/SubsetProperties.lean into several smaller files. Not only is it too huge, but the name is very unhelpful, since actually about 90% of the file is about compactness; I've moved this material into various files inside a new subdirectory Topology/Compactness/.

@@ -5,7 +5,7 @@ Authors: Yury Kudryashov
 -/
 import Mathlib.Topology.Instances.Irrational
 import Mathlib.Topology.Algebra.Order.Archimedean
-import Mathlib.Topology.Paracompact
+import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.MetricSpace.Metrizable
 import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Data.Set.Intervals.Monotone

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -286,7 +286,7 @@ theorem nhds_prod_antitone_basis_inv_pnat (x y : ℝₗ) :
 /-- The sets of rational and irrational points of the antidiagonal `{(x, y) | x + y = 0}` cannot be
 separated by open neighborhoods. This implies that `ℝₗ × ℝₗ` is not a normal space. -/
 theorem not_separatedNhds_rat_irrational_antidiag :
-  ¬SeparatedNhds {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ ∃ r : ℚ, ↑r = x.1}
+    ¬SeparatedNhds {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ ∃ r : ℚ, ↑r = x.1}
     {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ Irrational (toReal x.1)} := by
   have h₀ : ∀ {n : ℕ+}, 0 < (n : ℝ)⁻¹ := inv_pos.2 (Nat.cast_pos.2 (PNat.pos _))
   have h₀' : ∀ {n : ℕ+} {x : ℝ}, x < x + (n : ℝ)⁻¹ := lt_add_of_pos_right _ h₀

Later I'm going to split files like Lipschitz into 2: one in EMetricSpace/ and one in MetricSpace/.

@@ -7,7 +7,7 @@ import Mathlib.Topology.Instances.Irrational
 import Mathlib.Topology.Algebra.Order.Archimedean
 import Mathlib.Topology.Paracompact
 import Mathlib.Topology.MetricSpace.Metrizable
-import Mathlib.Topology.MetricSpace.EMetricParacompact
+import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module sorgenfrey_line
-! leanprover-community/mathlib commit 328375597f2c0dd00522d9c2e5a33b6a6128feeb
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Instances.Irrational
 import Mathlib.Topology.Algebra.Order.Archimedean
@@ -16,6 +11,8 @@ import Mathlib.Topology.MetricSpace.EMetricParacompact
 import Mathlib.Data.Set.Intervals.Monotone
 import Mathlib.Topology.Separation.NotNormal
 
+#align_import sorgenfrey_line from "leanprover-community/mathlib"@"328375597f2c0dd00522d9c2e5a33b6a6128feeb"
+
 /-!
 # Sorgenfrey line
 

@@ -14,6 +14,7 @@ import Mathlib.Topology.Paracompact
 import Mathlib.Topology.MetricSpace.Metrizable
 import Mathlib.Topology.MetricSpace.EMetricParacompact
 import Mathlib.Data.Set.Intervals.Monotone
+import Mathlib.Topology.Separation.NotNormal
 
 /-!
 # Sorgenfrey line
@@ -38,7 +39,7 @@ Prove that the Sorgenfrey line is a paracompact space.
 
 open Set Filter TopologicalSpace
 
-open scoped Topology Filter
+open scoped Topology Filter Cardinal
 
 namespace Counterexample
 
@@ -78,7 +79,7 @@ theorem isOpen_Ici (a : ℝₗ) : IsOpen (Ici a) :=
   iUnion_Ico_right a ▸ isOpen_iUnion (isOpen_Ico a)
 #align counterexample.sorgenfrey_line.is_open_Ici Counterexample.SorgenfreyLine.isOpen_Ici
 
-theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (fun b => a < b) fun b => Ico a b := by
+theorem nhds_basis_Ico (a : ℝₗ) : (𝓝 a).HasBasis (a < ·) (Ico a ·) := by
   rw [TopologicalSpace.nhds_generateFrom]
   haveI : Nonempty { x // x ≤ a } := Set.nonempty_Iic_subtype
   have : (⨅ x : { i // i ≤ a }, 𝓟 (Ici ↑x)) = 𝓟 (Ici a) := by
@@ -235,9 +236,15 @@ theorem isClopen_Ici_prod (x : ℝₗ × ℝₗ) : IsClopen (Ici x) :=
   (Ici_prod_eq x).symm ▸ (isClopen_Ici _).prod (isClopen_Ici _)
 #align counterexample.sorgenfrey_line.is_clopen_Ici_prod Counterexample.SorgenfreyLine.isClopen_Ici_prod
 
+theorem cardinal_antidiagonal (c : ℝₗ) : #{x : ℝₗ × ℝₗ | x.1 + x.2 = c} = 𝔠 := by
+  rw [← Cardinal.mk_real]
+  exact Equiv.cardinal_eq ⟨fun x ↦ toReal x.1.1,
+    fun x ↦ ⟨(toReal.symm x, c - toReal.symm x), by simp⟩,
+    fun ⟨x, hx⟩ ↦ by ext <;> simp [← hx.out], fun x ↦ rfl⟩
+
 /-- Any subset of an antidiagonal `{(x, y) : ℝₗ × ℝₗ| x + y = c}` is a closed set. -/
-theorem isClosed_of_subset_antidiagonal {s : Set (ℝₗ × ℝₗ)} {c : ℝₗ}
-    (hs : ∀ x : ℝₗ × ℝₗ, x ∈ s → x.1 + x.2 = c) : IsClosed s := by
+theorem isClosed_of_subset_antidiagonal {s : Set (ℝₗ × ℝₗ)} {c : ℝₗ} (hs : ∀ x ∈ s, x.1 + x.2 = c) :
+    IsClosed s := by
   rw [← closure_subset_iff_isClosed]
   rintro ⟨x, y⟩ H
   obtain rfl : x + y = c := by
@@ -252,30 +259,47 @@ theorem isClosed_of_subset_antidiagonal {s : Set (ℝₗ × ℝₗ)} {c : ℝₗ
     rwa [← add_le_add_iff_left, hs _ H, add_le_add_iff_right]
 #align counterexample.sorgenfrey_line.is_closed_of_subset_antidiagonal Counterexample.SorgenfreyLine.isClosed_of_subset_antidiagonal
 
+open Subtype in
+instance (c : ℝₗ) : DiscreteTopology {x : ℝₗ × ℝₗ | x.1 + x.2 = c} :=
+  forall_open_iff_discrete.1 fun U ↦ isClosed_compl_iff.1 <| isClosed_induced_iff.2
+    ⟨val '' Uᶜ, isClosed_of_subset_antidiagonal <| coe_image_subset _ Uᶜ,
+      preimage_image_eq _ val_injective⟩
+
+/-- The Sorgenfrey plane `ℝₗ × ℝₗ` is not a normal space. -/
+theorem not_normalSpace_prod : ¬NormalSpace (ℝₗ × ℝₗ) :=
+  (isClosed_antidiagonal 0).not_normal_of_continuum_le_mk (cardinal_antidiagonal _).ge
+#align counterexample.sorgenfrey_line.not_normal_space_prod Counterexample.SorgenfreyLine.not_normalSpace_prod
+
+/-- An antidiagonal is a separable set but is not a separable space. -/
+theorem isSeparable_antidiagonal (c : ℝₗ) : IsSeparable {x : ℝₗ × ℝₗ | x.1 + x.2 = c} :=
+  isSeparable_of_separableSpace _
+
+/-- An antidiagonal is a separable set but is not a separable space. -/
+theorem not_separableSpace_antidiagonal (c : ℝₗ) :
+    ¬SeparableSpace {x : ℝₗ × ℝₗ | x.1 + x.2 = c} := by
+  rw [separableSpace_iff_countable, ← Cardinal.mk_le_aleph0_iff, cardinal_antidiagonal, not_le]
+  exact Cardinal.aleph0_lt_continuum
+
 theorem nhds_prod_antitone_basis_inv_pnat (x y : ℝₗ) :
     (𝓝 (x, y)).HasAntitoneBasis fun n : ℕ+ => Ico x (x + (n : ℝₗ)⁻¹) ×ˢ Ico y (y + (n : ℝₗ)⁻¹) := by
   rw [nhds_prod_eq]
   exact (nhds_antitone_basis_Ico_inv_pnat x).prod (nhds_antitone_basis_Ico_inv_pnat y)
 #align counterexample.sorgenfrey_line.nhds_prod_antitone_basis_inv_pnat Counterexample.SorgenfreyLine.nhds_prod_antitone_basis_inv_pnat
 
-/-- The product of the Sorgenfrey line and itself is not a normal topological space. -/
-theorem not_normalSpace_prod : ¬NormalSpace (ℝₗ × ℝₗ) := by
+/-- The sets of rational and irrational points of the antidiagonal `{(x, y) | x + y = 0}` cannot be
+separated by open neighborhoods. This implies that `ℝₗ × ℝₗ` is not a normal space. -/
+theorem not_separatedNhds_rat_irrational_antidiag :
+  ¬SeparatedNhds {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ ∃ r : ℚ, ↑r = x.1}
+    {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ Irrational (toReal x.1)} := by
   have h₀ : ∀ {n : ℕ+}, 0 < (n : ℝ)⁻¹ := inv_pos.2 (Nat.cast_pos.2 (PNat.pos _))
   have h₀' : ∀ {n : ℕ+} {x : ℝ}, x < x + (n : ℝ)⁻¹ := lt_add_of_pos_right _ h₀
-  intro
   /- Let `S` be the set of points `(x, y)` on the line `x + y = 0` such that `x` is rational.
     Let `T` be the set of points `(x, y)` on the line `x + y = 0` such that `x` is irrational.
     These sets are closed, see `SorgenfreyLine.isClosed_of_subset_antidiagonal`, and disjoint. -/
   set S := {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ ∃ r : ℚ, ↑r = x.1}
   set T := {x : ℝₗ × ℝₗ | x.1 + x.2 = 0 ∧ Irrational (toReal x.1)}
-  have hSc : IsClosed S := isClosed_of_subset_antidiagonal fun x hx => hx.1
-  have hTc : IsClosed T := isClosed_of_subset_antidiagonal fun x hx => hx.1
-  have hd : Disjoint S T := by
-    rw [disjoint_iff_inf_le]
-    rintro ⟨x, y⟩ ⟨⟨-, r, rfl : _ = x⟩, -, hr⟩
-    exact r.not_irrational hr
   -- Consider disjoint open sets `U ⊇ S` and `V ⊇ T`.
-  rcases normal_separation hSc hTc hd with ⟨U, V, Uo, Vo, SU, TV, UV⟩
+  rintro ⟨U, V, Uo, Vo, SU, TV, UV⟩
   /- For each point `(x, -x) ∈ T`, choose a neighborhood
     `Ico x (x + k⁻¹) ×ˢ Ico (-x) (-x + k⁻¹) ⊆ V`. -/
   have : ∀ x : ℝₗ, Irrational (toReal x) →
@@ -316,7 +340,6 @@ theorem not_normalSpace_prod : ¬NormalSpace (ℝₗ × ℝₗ) := by
   · refine' (nhds_antitone_basis_Ico_inv_pnat (-x)).2 hnN ⟨neg_le_neg hxn.1.le, _⟩
     simp only [add_neg_lt_iff_le_add', lt_neg_add_iff_add_lt]
     exact hxn.2
-#align counterexample.sorgenfrey_line.not_normal_space_prod Counterexample.SorgenfreyLine.not_normalSpace_prod
 
 /-- Topology on the Sorgenfrey line is not metrizable. -/
 theorem not_metrizableSpace : ¬MetrizableSpace ℝₗ := by

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>