topology.order.lower_topology ⟷ Mathlib.Topology.Order.LowerUpperTopology

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

@@ -238,7 +238,7 @@ theorem Topology.IsLower.isClosed_upperClosure (h : s.Finite) : IsClosed (upperC
 /-- Every set open in the lower topology is a lower set. -/
 theorem Topology.IsLower.isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
-  rw [is_open_iff_generate_Ici_compl] at h 
+  rw [is_open_iff_generate_Ici_compl] at h
   induction h
   case basic u h => obtain ⟨a, rfl⟩ := h; exact (isUpperSet_Ici a).compl
   case univ => exact isLowerSet_univ
@@ -278,7 +278,7 @@ protected theorem Topology.IsLower.isTopologicalBasis :
     rw [sInter_image]
   · rintro ⟨F, ⟨hF, hs⟩, rfl⟩
     haveI := hF.to_subtype
-    rw [subset_def, Subtype.forall'] at hs 
+    rw [subset_def, Subtype.forall'] at hs
     choose f hf using hs
     exact ⟨_, finite_range f, by simp_rw [bInter_range, hf, sInter_eq_Inter]⟩
 #align lower_topology.is_topological_basis Topology.IsLower.isTopologicalBasis

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

@@ -198,15 +198,15 @@ section Preorder
 
 variable [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α}
 
-#print Topology.IsLower.WithLowerHomeomorph /-
+#print Topology.IsLower.withLowerHomeomorph /-
 /-- If `α` is equipped with the lower topology, then it is homeomorphic to `with_lower_topology α`.
 -/
-def Topology.IsLower.WithLowerHomeomorph : Topology.WithLower α ≃ₜ α :=
+def Topology.IsLower.withLowerHomeomorph : Topology.WithLower α ≃ₜ α :=
   {
     Topology.WithLower.ofLower with
     continuous_toFun := by convert continuous_id; apply topology_eq_lower_topology
     continuous_invFun := by convert ← continuous_id; apply topology_eq_lower_topology }
-#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.WithLowerHomeomorph
+#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.withLowerHomeomorph
 -/
 
 #print Topology.IsLower.isOpen_iff_generate_Ici_compl /-

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

@@ -172,7 +172,7 @@ theorem Topology.WithLower.isOpen_def (T : Set (Topology.WithLower α)) :
 end Topology.WithLower
 
 #print Topology.IsLower /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`topology_eq_isLower] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`topology_eq_isLower] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/

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

@@ -172,7 +172,7 @@ theorem Topology.WithLower.isOpen_def (T : Set (Topology.WithLower α)) :
 end Topology.WithLower
 
 #print Topology.IsLower /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_isLower] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`topology_eq_isLower] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/

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

@@ -55,157 +55,164 @@ variable (α β : Type _)
 
 open Set TopologicalSpace
 
-#print WithLowerTopology /-
+#print Topology.WithLower /-
 /-- Type synonym for a preorder equipped with the lower topology
 -/
-def WithLowerTopology :=
+def Topology.WithLower :=
   α
-#align with_lower_topology WithLowerTopology
+#align with_lower_topology Topology.WithLower
 -/
 
 variable {α β}
 
-namespace WithLowerTopology
+namespace Topology.WithLower
 
-#print WithLowerTopology.toLower /-
+#print Topology.WithLower.toLower /-
 /-- `to_lower` is the identity function to the `with_lower_topology` of a type.  -/
 @[match_pattern]
-def toLower : α ≃ WithLowerTopology α :=
+def Topology.WithLower.toLower : α ≃ Topology.WithLower α :=
   Equiv.refl _
-#align with_lower_topology.to_lower WithLowerTopology.toLower
+#align with_lower_topology.to_lower Topology.WithLower.toLower
 -/
 
-#print WithLowerTopology.ofLower /-
+#print Topology.WithLower.ofLower /-
 /-- `of_lower` is the identity function from the `with_lower_topology` of a type.  -/
 @[match_pattern]
-def ofLower : WithLowerTopology α ≃ α :=
+def Topology.WithLower.ofLower : Topology.WithLower α ≃ α :=
   Equiv.refl _
-#align with_lower_topology.of_lower WithLowerTopology.ofLower
+#align with_lower_topology.of_lower Topology.WithLower.ofLower
 -/
 
-#print WithLowerTopology.to_withLowerTopology_symm_eq /-
+#print Topology.WithLower.to_WithLower_symm_eq /-
 @[simp]
-theorem to_withLowerTopology_symm_eq : (@toLower α).symm = ofLower :=
+theorem Topology.WithLower.to_WithLower_symm_eq :
+    (@Topology.WithLower.toLower α).symm = Topology.WithLower.ofLower :=
   rfl
-#align with_lower_topology.to_with_lower_topology_symm_eq WithLowerTopology.to_withLowerTopology_symm_eq
+#align with_lower_topology.to_with_lower_topology_symm_eq Topology.WithLower.to_WithLower_symm_eq
 -/
 
-#print WithLowerTopology.of_withLowerTopology_symm_eq /-
+#print Topology.WithLower.of_WithLower_symm_eq /-
 @[simp]
-theorem of_withLowerTopology_symm_eq : (@ofLower α).symm = toLower :=
+theorem Topology.WithLower.of_WithLower_symm_eq :
+    (@Topology.WithLower.ofLower α).symm = Topology.WithLower.toLower :=
   rfl
-#align with_lower_topology.of_with_lower_topology_symm_eq WithLowerTopology.of_withLowerTopology_symm_eq
+#align with_lower_topology.of_with_lower_topology_symm_eq Topology.WithLower.of_WithLower_symm_eq
 -/
 
-#print WithLowerTopology.toLower_ofLower /-
+#print Topology.WithLower.toLower_ofLower /-
 @[simp]
-theorem toLower_ofLower (a : WithLowerTopology α) : toLower (ofLower a) = a :=
+theorem Topology.WithLower.toLower_ofLower (a : Topology.WithLower α) :
+    Topology.WithLower.toLower (Topology.WithLower.ofLower a) = a :=
   rfl
-#align with_lower_topology.to_lower_of_lower WithLowerTopology.toLower_ofLower
+#align with_lower_topology.to_lower_of_lower Topology.WithLower.toLower_ofLower
 -/
 
-#print WithLowerTopology.ofLower_toLower /-
+#print Topology.WithLower.ofLower_toLower /-
 @[simp]
-theorem ofLower_toLower (a : α) : ofLower (toLower a) = a :=
+theorem Topology.WithLower.ofLower_toLower (a : α) :
+    Topology.WithLower.ofLower (Topology.WithLower.toLower a) = a :=
   rfl
-#align with_lower_topology.of_lower_to_lower WithLowerTopology.ofLower_toLower
+#align with_lower_topology.of_lower_to_lower Topology.WithLower.ofLower_toLower
 -/
 
-#print WithLowerTopology.toLower_inj /-
+#print Topology.WithLower.toLower_inj /-
 @[simp]
-theorem toLower_inj {a b : α} : toLower a = toLower b ↔ a = b :=
+theorem Topology.WithLower.toLower_inj {a b : α} :
+    Topology.WithLower.toLower a = Topology.WithLower.toLower b ↔ a = b :=
   Iff.rfl
-#align with_lower_topology.to_lower_inj WithLowerTopology.toLower_inj
+#align with_lower_topology.to_lower_inj Topology.WithLower.toLower_inj
 -/
 
-#print WithLowerTopology.ofLower_inj /-
+#print Topology.WithLower.ofLower_inj /-
 @[simp]
-theorem ofLower_inj {a b : WithLowerTopology α} : ofLower a = ofLower b ↔ a = b :=
+theorem Topology.WithLower.ofLower_inj {a b : Topology.WithLower α} :
+    Topology.WithLower.ofLower a = Topology.WithLower.ofLower b ↔ a = b :=
   Iff.rfl
-#align with_lower_topology.of_lower_inj WithLowerTopology.ofLower_inj
+#align with_lower_topology.of_lower_inj Topology.WithLower.ofLower_inj
 -/
 
-#print WithLowerTopology.rec /-
+#print Topology.WithLower.rec /-
 /-- A recursor for `with_lower_topology`. Use as `induction x using with_lower_topology.rec`. -/
-protected def rec {β : WithLowerTopology α → Sort _} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
-  h (ofLower a)
-#align with_lower_topology.rec WithLowerTopology.rec
+protected def Topology.WithLower.rec {β : Topology.WithLower α → Sort _}
+    (h : ∀ a, β (Topology.WithLower.toLower a)) : ∀ a, β a := fun a =>
+  h (Topology.WithLower.ofLower a)
+#align with_lower_topology.rec Topology.WithLower.rec
 -/
 
-instance [Nonempty α] : Nonempty (WithLowerTopology α) :=
+instance [Nonempty α] : Nonempty (Topology.WithLower α) :=
   ‹Nonempty α›
 
-instance [Inhabited α] : Inhabited (WithLowerTopology α) :=
+instance [Inhabited α] : Inhabited (Topology.WithLower α) :=
   ‹Inhabited α›
 
 variable [Preorder α]
 
-instance : Preorder (WithLowerTopology α) :=
+instance : Preorder (Topology.WithLower α) :=
   ‹Preorder α›
 
-instance : TopologicalSpace (WithLowerTopology α) :=
+instance : TopologicalSpace (Topology.WithLower α) :=
   generateFrom {s | ∃ a, Ici aᶜ = s}
 
-#print WithLowerTopology.isOpen_preimage_ofLower /-
-theorem isOpen_preimage_ofLower (S : Set α) :
-    IsOpen (WithLowerTopology.ofLower ⁻¹' S) ↔
+#print Topology.WithLower.isOpen_preimage_ofLower /-
+theorem Topology.WithLower.isOpen_preimage_ofLower (S : Set α) :
+    IsOpen (Topology.WithLower.ofLower ⁻¹' S) ↔
       (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen S :=
   Iff.rfl
-#align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
+#align with_lower_topology.is_open_preimage_of_lower Topology.WithLower.isOpen_preimage_ofLower
 -/
 
-#print WithLowerTopology.isOpen_def /-
-theorem isOpen_def (T : Set (WithLowerTopology α)) :
+#print Topology.WithLower.isOpen_def /-
+theorem Topology.WithLower.isOpen_def (T : Set (Topology.WithLower α)) :
     IsOpen T ↔
-      (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
+      (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen (Topology.WithLower.toLower ⁻¹' T) :=
   Iff.rfl
-#align with_lower_topology.is_open_def WithLowerTopology.isOpen_def
+#align with_lower_topology.is_open_def Topology.WithLower.isOpen_def
 -/
 
-end WithLowerTopology
+end Topology.WithLower
 
-#print LowerTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
+#print Topology.IsLower /-
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_isLower] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/
-class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
-  topology_eq_lowerTopology : t = generateFrom {s | ∃ a, Ici aᶜ = s}
-#align lower_topology LowerTopology
+class Topology.IsLower (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
+  topology_eq_isLower : t = generateFrom {s | ∃ a, Ici aᶜ = s}
+#align lower_topology Topology.IsLower
 -/
 
-instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
+instance [Preorder α] : Topology.IsLower (Topology.WithLower α) :=
   ⟨rfl⟩
 
-namespace LowerTopology
+namespace Topology.IsLower
 
-#print LowerTopology.lowerBasis /-
+#print Topology.IsLower.lowerBasis /-
 /-- The complements of the upper closures of finite sets are a collection of lower sets
 which form a basis for the lower topology. -/
-def lowerBasis (α : Type _) [Preorder α] :=
+def Topology.IsLower.lowerBasis (α : Type _) [Preorder α] :=
   {s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s}
-#align lower_topology.lower_basis LowerTopology.lowerBasis
+#align lower_topology.lower_basis Topology.IsLower.lowerBasis
 -/
 
 section Preorder
 
-variable [Preorder α] [TopologicalSpace α] [LowerTopology α] {s : Set α}
+variable [Preorder α] [TopologicalSpace α] [Topology.IsLower α] {s : Set α}
 
-#print LowerTopology.withLowerTopologyHomeomorph /-
+#print Topology.IsLower.WithLowerHomeomorph /-
 /-- If `α` is equipped with the lower topology, then it is homeomorphic to `with_lower_topology α`.
 -/
-def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
+def Topology.IsLower.WithLowerHomeomorph : Topology.WithLower α ≃ₜ α :=
   {
-    WithLowerTopology.ofLower with
+    Topology.WithLower.ofLower with
     continuous_toFun := by convert continuous_id; apply topology_eq_lower_topology
     continuous_invFun := by convert ← continuous_id; apply topology_eq_lower_topology }
-#align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
+#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.WithLowerHomeomorph
 -/
 
-#print LowerTopology.isOpen_iff_generate_Ici_compl /-
-theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s := by
-  rw [topology_eq_lower_topology α] <;> rfl
-#align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
+#print Topology.IsLower.isOpen_iff_generate_Ici_compl /-
+theorem Topology.IsLower.isOpen_iff_generate_Ici_compl :
+    IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s := by rw [topology_eq_lower_topology α] <;> rfl
+#align lower_topology.is_open_iff_generate_Ici_compl Topology.IsLower.isOpen_iff_generate_Ici_compl
 -/
 
 /- warning: lower_topology.is_closed_Ici clashes with is_closed_Ici -> isClosed_Ici
@@ -213,22 +220,23 @@ Case conversion may be inaccurate. Consider using '#align lower_topology.is_clos
 #print isClosed_Ici /-
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
 theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
-  isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
+  isOpen_compl_iff.1 <|
+    Topology.IsLower.isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
 #align lower_topology.is_closed_Ici isClosed_Ici
 -/
 
-#print LowerTopology.isClosed_upperClosure /-
+#print Topology.IsLower.isClosed_upperClosure /-
 /-- The upper closure of a finite set is closed in the lower topology. -/
-theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) :=
+theorem Topology.IsLower.isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) :=
   by
   simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
   exact Set.Finite.isClosed_biUnion h fun a h₁ => isClosed_Ici a
-#align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
+#align lower_topology.is_closed_upper_closure Topology.IsLower.isClosed_upperClosure
 -/
 
-#print LowerTopology.isLowerSet_of_isOpen /-
+#print Topology.IsLower.isLowerSet_of_isOpen /-
 /-- Every set open in the lower topology is a lower set. -/
-theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
+theorem Topology.IsLower.isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
   rw [is_open_iff_generate_Ici_compl] at h 
   induction h
@@ -236,29 +244,30 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   case univ => exact isLowerSet_univ
   case inter u v hu1 hv1 hu2 hv2 => exact hu2.inter hv2
   case sUnion _ _ ih => exact isLowerSet_sUnion ih
-#align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
+#align lower_topology.is_lower_set_of_is_open Topology.IsLower.isLowerSet_of_isOpen
 -/
 
-#print LowerTopology.isUpperSet_of_isClosed /-
-theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
-  isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl
-#align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosed
+#print Topology.IsLower.isUpperSet_of_isClosed /-
+theorem Topology.IsLower.isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
+  isLowerSet_compl.1 <| Topology.IsLower.isLowerSet_of_isOpen h.isOpen_compl
+#align lower_topology.is_upper_set_of_is_closed Topology.IsLower.isUpperSet_of_isClosed
 -/
 
-#print LowerTopology.closure_singleton /-
+#print Topology.IsLower.closure_singleton /-
 /--
 The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval
 [a, ∞).
 -/
 @[simp]
-theorem closure_singleton (a : α) : closure {a} = Ici a :=
+theorem Topology.IsLower.closure_singleton (a : α) : closure {a} = Ici a :=
   subset_antisymm ((closure_minimal fun b h => h.ge) <| isClosed_Ici a) <|
-    (isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl
-#align lower_topology.closure_singleton LowerTopology.closure_singleton
+    (Topology.IsLower.isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl
+#align lower_topology.closure_singleton Topology.IsLower.closure_singleton
 -/
 
-#print LowerTopology.isTopologicalBasis /-
-protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) :=
+#print Topology.IsLower.isTopologicalBasis /-
+protected theorem Topology.IsLower.isTopologicalBasis :
+    IsTopologicalBasis (Topology.IsLower.lowerBasis α) :=
   by
   convert is_topological_basis_of_subbasis (topology_eq_lower_topology α)
   simp_rw [lower_basis, coe_upperClosure, compl_Union]
@@ -272,14 +281,14 @@ protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) :=
     rw [subset_def, Subtype.forall'] at hs 
     choose f hf using hs
     exact ⟨_, finite_range f, by simp_rw [bInter_range, hf, sInter_eq_Inter]⟩
-#align lower_topology.is_topological_basis LowerTopology.isTopologicalBasis
+#align lower_topology.is_topological_basis Topology.IsLower.isTopologicalBasis
 -/
 
 end Preorder
 
 section PartialOrder
 
-variable [PartialOrder α] [TopologicalSpace α] [LowerTopology α]
+variable [PartialOrder α] [TopologicalSpace α] [Topology.IsLower α]
 
 -- see Note [lower instance priority]
 /-- The lower topology on a partial order is T₀.
@@ -290,16 +299,17 @@ instance (priority := 90) : T0Space α :=
 
 end PartialOrder
 
-end LowerTopology
+end Topology.IsLower
 
-instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [Preorder β]
-    [TopologicalSpace β] [LowerTopology β] [OrderBot β] : LowerTopology (α × β)
-    where topology_eq_lowerTopology :=
+instance [Preorder α] [TopologicalSpace α] [Topology.IsLower α] [OrderBot α] [Preorder β]
+    [TopologicalSpace β] [Topology.IsLower β] [OrderBot β] : Topology.IsLower (α × β)
+    where topology_eq_isLower :=
     by
     refine' le_antisymm (le_generateFrom _) _
     · rintro _ ⟨x, rfl⟩
       exact ((isClosed_Ici _).Prod <| isClosed_Ici _).isOpen_compl
-    rw [(lower_topology.is_topological_basis.prod LowerTopology.isTopologicalBasis).eq_generateFrom,
+    rw [(lower_topology.is_topological_basis.prod
+          Topology.IsLower.isTopologicalBasis).eq_generateFrom,
       le_generate_from_iff_subset_is_open, image2_subset_iff]
     rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
     dsimp
@@ -313,14 +323,14 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
 
 section CompleteLattice
 
-variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
-  [TopologicalSpace β] [LowerTopology β]
+variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [Topology.IsLower α]
+  [TopologicalSpace β] [Topology.IsLower β]
 
 #print sInfHom.continuous /-
 theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   by
   convert continuous_generateFrom _
-  · exact LowerTopology.topology_eq_lowerTopology β
+  · exact Topology.IsLower.topology_eq_isLower β
   rintro _ ⟨b, rfl⟩
   rw [preimage_compl, isOpen_compl_iff]
   convert isClosed_Ici (Inf <| f ⁻¹' Ici b)
@@ -329,11 +339,11 @@ theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
 #align Inf_hom.continuous sInfHom.continuous
 -/
 
-#print LowerTopology.continuousInf /-
+#print Topology.IsLower.toContinuousInf /-
 -- see Note [lower instance priority]
-instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
+instance (priority := 90) Topology.IsLower.toContinuousInf : ContinuousInf α :=
   ⟨(infsInfHom : sInfHom (α × α) α).Continuous⟩
-#align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
+#align lower_topology.to_has_continuous_inf Topology.IsLower.toContinuousInf
 -/
 
 end CompleteLattice

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

@@ -3,9 +3,9 @@ Copyright (c) 2023 Christopher Hoskin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
 -/
-import Mathbin.Topology.Homeomorph
-import Mathbin.Topology.Order.Lattice
-import Mathbin.Order.Hom.CompleteLattice
+import Topology.Homeomorph
+import Topology.Order.Lattice
+import Order.Hom.CompleteLattice
 
 #align_import topology.order.lower_topology from "leanprover-community/mathlib"@"34ee86e6a59d911a8e4f89b68793ee7577ae79c7"
 
@@ -165,7 +165,7 @@ theorem isOpen_def (T : Set (WithLowerTopology α)) :
 end WithLowerTopology
 
 #print LowerTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -222,7 +222,7 @@ theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) :=
   by
   simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
-  exact isClosed_biUnion h fun a h₁ => isClosed_Ici a
+  exact Set.Finite.isClosed_biUnion h fun a h₁ => isClosed_Ici a
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 -/
 
@@ -305,7 +305,8 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
     dsimp
     simp_rw [coe_upperClosure, compl_Union, prod_eq, preimage_Inter, preimage_compl]
     -- Note: `refine` doesn't work here because it tries using `prod.topological_space`.
-    apply (isOpen_biInter hs fun a _ => _).inter (isOpen_biInter ht fun b _ => _)
+    apply
+      (Set.Finite.isOpen_biInter hs fun a _ => _).inter (Set.Finite.isOpen_biInter ht fun b _ => _)
     · exact generate_open.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact generate_open.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
     all_goals infer_instance

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -208,11 +208,13 @@ theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen {t | ∃ a, Ic
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
 -/
 
-#print LowerTopology.isClosed_Ici /-
+/- warning: lower_topology.is_closed_Ici clashes with is_closed_Ici -> isClosed_Ici
+Case conversion may be inaccurate. Consider using '#align lower_topology.is_closed_Ici isClosed_Iciₓ'. -/
+#print isClosed_Ici /-
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
 theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
   isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
-#align lower_topology.is_closed_Ici LowerTopology.isClosed_Ici
+#align lower_topology.is_closed_Ici isClosed_Ici
 -/
 
 #print LowerTopology.isClosed_upperClosure /-
@@ -296,7 +298,7 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
     by
     refine' le_antisymm (le_generateFrom _) _
     · rintro _ ⟨x, rfl⟩
-      exact ((LowerTopology.isClosed_Ici _).Prod <| LowerTopology.isClosed_Ici _).isOpen_compl
+      exact ((isClosed_Ici _).Prod <| isClosed_Ici _).isOpen_compl
     rw [(lower_topology.is_topological_basis.prod LowerTopology.isTopologicalBasis).eq_generateFrom,
       le_generate_from_iff_subset_is_open, image2_subset_iff]
     rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
@@ -320,7 +322,7 @@ theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   · exact LowerTopology.topology_eq_lowerTopology β
   rintro _ ⟨b, rfl⟩
   rw [preimage_compl, isOpen_compl_iff]
-  convert LowerTopology.isClosed_Ici (Inf <| f ⁻¹' Ici b)
+  convert isClosed_Ici (Inf <| f ⁻¹' Ici b)
   refine' subset_antisymm (fun a => sInf_le) fun a ha => le_trans _ <| OrderHomClass.mono f ha
   simp [map_Inf]
 #align Inf_hom.continuous sInfHom.continuous

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

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Christopher Hoskin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
-
-! This file was ported from Lean 3 source module topology.order.lower_topology
-! leanprover-community/mathlib commit 34ee86e6a59d911a8e4f89b68793ee7577ae79c7
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Homeomorph
 import Mathbin.Topology.Order.Lattice
 import Mathbin.Order.Hom.CompleteLattice
 
+#align_import topology.order.lower_topology from "leanprover-community/mathlib"@"34ee86e6a59d911a8e4f89b68793ee7577ae79c7"
+
 /-!
 # Lower topology
 

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

@@ -157,16 +157,18 @@ theorem isOpen_preimage_ofLower (S : Set α) :
 #align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
 -/
 
+#print WithLowerTopology.isOpen_def /-
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
     IsOpen T ↔
       (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
   Iff.rfl
 #align with_lower_topology.is_open_def WithLowerTopology.isOpen_def
+-/
 
 end WithLowerTopology
 
 #print LowerTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/
@@ -192,6 +194,7 @@ section Preorder
 
 variable [Preorder α] [TopologicalSpace α] [LowerTopology α] {s : Set α}
 
+#print LowerTopology.withLowerTopologyHomeomorph /-
 /-- If `α` is equipped with the lower topology, then it is homeomorphic to `with_lower_topology α`.
 -/
 def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
@@ -200,10 +203,13 @@ def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
     continuous_toFun := by convert continuous_id; apply topology_eq_lower_topology
     continuous_invFun := by convert ← continuous_id; apply topology_eq_lower_topology }
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
+-/
 
+#print LowerTopology.isOpen_iff_generate_Ici_compl /-
 theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s := by
   rw [topology_eq_lower_topology α] <;> rfl
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
+-/
 
 #print LowerTopology.isClosed_Ici /-
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
@@ -310,6 +316,7 @@ section CompleteLattice
 variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
   [TopologicalSpace β] [LowerTopology β]
 
+#print sInfHom.continuous /-
 theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   by
   convert continuous_generateFrom _
@@ -320,11 +327,14 @@ theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   refine' subset_antisymm (fun a => sInf_le) fun a ha => le_trans _ <| OrderHomClass.mono f ha
   simp [map_Inf]
 #align Inf_hom.continuous sInfHom.continuous
+-/
 
+#print LowerTopology.continuousInf /-
 -- see Note [lower instance priority]
 instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
   ⟨(infsInfHom : sInfHom (α × α) α).Continuous⟩
 #align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
+-/
 
 end CompleteLattice
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -147,31 +147,31 @@ instance : Preorder (WithLowerTopology α) :=
   ‹Preorder α›
 
 instance : TopologicalSpace (WithLowerTopology α) :=
-  generateFrom { s | ∃ a, Ici aᶜ = s }
+  generateFrom {s | ∃ a, Ici aᶜ = s}
 
 #print WithLowerTopology.isOpen_preimage_ofLower /-
 theorem isOpen_preimage_ofLower (S : Set α) :
     IsOpen (WithLowerTopology.ofLower ⁻¹' S) ↔
-      (generateFrom { s : Set α | ∃ a : α, Ici aᶜ = s }).IsOpen S :=
+      (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen S :=
   Iff.rfl
 #align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
 -/
 
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
     IsOpen T ↔
-      (generateFrom { s : Set α | ∃ a : α, Ici aᶜ = s }).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
+      (generateFrom {s : Set α | ∃ a : α, Ici aᶜ = s}).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
   Iff.rfl
 #align with_lower_topology.is_open_def WithLowerTopology.isOpen_def
 
 end WithLowerTopology
 
 #print LowerTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/
 class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
-  topology_eq_lowerTopology : t = generateFrom { s | ∃ a, Ici aᶜ = s }
+  topology_eq_lowerTopology : t = generateFrom {s | ∃ a, Ici aᶜ = s}
 #align lower_topology LowerTopology
 -/
 
@@ -184,7 +184,7 @@ namespace LowerTopology
 /-- The complements of the upper closures of finite sets are a collection of lower sets
 which form a basis for the lower topology. -/
 def lowerBasis (α : Type _) [Preorder α] :=
-  { s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s }
+  {s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s}
 #align lower_topology.lower_basis LowerTopology.lowerBasis
 -/
 
@@ -198,10 +198,10 @@ def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
   {
     WithLowerTopology.ofLower with
     continuous_toFun := by convert continuous_id; apply topology_eq_lower_topology
-    continuous_invFun := by convert← continuous_id; apply topology_eq_lower_topology }
+    continuous_invFun := by convert ← continuous_id; apply topology_eq_lower_topology }
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
 
-theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, Ici aᶜ = t } s := by
+theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s := by
   rw [topology_eq_lower_topology α] <;> rfl
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -225,7 +225,7 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
-  rw [is_open_iff_generate_Ici_compl] at h
+  rw [is_open_iff_generate_Ici_compl] at h 
   induction h
   case basic u h => obtain ⟨a, rfl⟩ := h; exact (isUpperSet_Ici a).compl
   case univ => exact isLowerSet_univ
@@ -264,7 +264,7 @@ protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) :=
     rw [sInter_image]
   · rintro ⟨F, ⟨hF, hs⟩, rfl⟩
     haveI := hF.to_subtype
-    rw [subset_def, Subtype.forall'] at hs
+    rw [subset_def, Subtype.forall'] at hs 
     choose f hf using hs
     exact ⟨_, finite_range f, by simp_rw [bInter_range, hf, sInter_eq_Inter]⟩
 #align lower_topology.is_topological_basis LowerTopology.isTopologicalBasis

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -221,6 +221,7 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 -/
 
+#print LowerTopology.isLowerSet_of_isOpen /-
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
@@ -231,10 +232,13 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   case inter u v hu1 hv1 hu2 hv2 => exact hu2.inter hv2
   case sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
+-/
 
+#print LowerTopology.isUpperSet_of_isClosed /-
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
   isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl
 #align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosed
+-/
 
 #print LowerTopology.closure_singleton /-
 /--

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -157,12 +157,6 @@ theorem isOpen_preimage_ofLower (S : Set α) :
 #align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
 -/
 
-/- warning: with_lower_topology.is_open_def -> WithLowerTopology.isOpen_def is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.topologicalSpace.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) => α -> (WithLowerTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => WithLowerTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
-Case conversion may be inaccurate. Consider using '#align with_lower_topology.is_open_def WithLowerTopology.isOpen_defₓ'. -/
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
     IsOpen T ↔
       (generateFrom { s : Set α | ∃ a : α, Ici aᶜ = s }).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
@@ -198,12 +192,6 @@ section Preorder
 
 variable [Preorder α] [TopologicalSpace α] [LowerTopology α] {s : Set α}
 
-/- warning: lower_topology.with_lower_topology_homeomorph -> LowerTopology.withLowerTopologyHomeomorph is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1], Homeomorph.{u1, u1} (WithLowerTopology.{u1} α) α (WithLowerTopology.topologicalSpace.{u1} α _inst_1) _inst_2
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1], Homeomorph.{u1, u1} (WithLowerTopology.{u1} α) α (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) _inst_2
-Case conversion may be inaccurate. Consider using '#align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorphₓ'. -/
 /-- If `α` is equipped with the lower topology, then it is homeomorphic to `with_lower_topology α`.
 -/
 def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
@@ -213,12 +201,6 @@ def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
     continuous_invFun := by convert← continuous_id; apply topology_eq_lower_topology }
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
 
-/- warning: lower_topology.is_open_iff_generate_Ici_compl -> LowerTopology.isOpen_iff_generate_Ici_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_2 s) (TopologicalSpace.GenerateOpen.{u1} α (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) t))) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_2 s) (TopologicalSpace.GenerateOpen.{u1} α (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) t))) s)
-Case conversion may be inaccurate. Consider using '#align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_complₓ'. -/
 theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, Ici aᶜ = t } s := by
   rw [topology_eq_lower_topology α] <;> rfl
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
@@ -239,12 +221,6 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 -/
 
-/- warning: lower_topology.is_lower_set_of_is_open -> LowerTopology.isLowerSet_of_isOpen is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsOpen.{u1} α _inst_2 s) -> (IsLowerSet.{u1} α (Preorder.toHasLe.{u1} α _inst_1) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsOpen.{u1} α _inst_2 s) -> (IsLowerSet.{u1} α (Preorder.toLE.{u1} α _inst_1) s)
-Case conversion may be inaccurate. Consider using '#align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpenₓ'. -/
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
@@ -256,12 +232,6 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   case sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
 
-/- warning: lower_topology.is_upper_set_of_is_closed -> LowerTopology.isUpperSet_of_isClosed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsClosed.{u1} α _inst_2 s) -> (IsUpperSet.{u1} α (Preorder.toHasLe.{u1} α _inst_1) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsClosed.{u1} α _inst_2 s) -> (IsUpperSet.{u1} α (Preorder.toLE.{u1} α _inst_1) s)
-Case conversion may be inaccurate. Consider using '#align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosedₓ'. -/
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
   isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl
 #align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosed
@@ -336,12 +306,6 @@ section CompleteLattice
 variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
   [TopologicalSpace β] [LowerTopology β]
 
-/- warning: Inf_hom.continuous -> sInfHom.continuous is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : CompleteLattice.{u2} β] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : LowerTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_2)))] (f : sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))), Continuous.{u1, u2} α β _inst_3 _inst_5 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) (fun (_x : sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) => α -> β) (sInfHom.hasCoeToFun.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : CompleteLattice.{u1} β] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : LowerTopology.{u2} α _inst_3 (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : LowerTopology.{u1} β _inst_5 (PartialOrder.toPreorder.{u1} β (CompleteSemilatticeInf.toPartialOrder.{u1} β (CompleteLattice.toCompleteSemilatticeInf.{u1} β _inst_2)))] (f : sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))), Continuous.{u2, u1} α β _inst_3 _inst_5 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.374 : α) => β) _x) (sInfHomClass.toFunLike.{max u2 u1, u2, u1} (sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)) (sInfHom.instSInfHomClassSInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)))) f)
-Case conversion may be inaccurate. Consider using '#align Inf_hom.continuous sInfHom.continuousₓ'. -/
 theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   by
   convert continuous_generateFrom _
@@ -353,12 +317,6 @@ theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   simp [map_Inf]
 #align Inf_hom.continuous sInfHom.continuous
 
-/- warning: lower_topology.to_has_continuous_inf -> LowerTopology.continuousInf is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))], ContinuousInf.{u1} α _inst_3 (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))], ContinuousInf.{u1} α _inst_3 (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))
-Case conversion may be inaccurate. Consider using '#align lower_topology.to_has_continuous_inf LowerTopology.continuousInfₓ'. -/
 -- see Note [lower instance priority]
 instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
   ⟨(infsInfHom : sInfHom (α × α) α).Continuous⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -209,12 +209,8 @@ Case conversion may be inaccurate. Consider using '#align lower_topology.with_lo
 def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
   {
     WithLowerTopology.ofLower with
-    continuous_toFun := by
-      convert continuous_id
-      apply topology_eq_lower_topology
-    continuous_invFun := by
-      convert← continuous_id
-      apply topology_eq_lower_topology }
+    continuous_toFun := by convert continuous_id; apply topology_eq_lower_topology
+    continuous_invFun := by convert← continuous_id; apply topology_eq_lower_topology }
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
 
 /- warning: lower_topology.is_open_iff_generate_Ici_compl -> LowerTopology.isOpen_iff_generate_Ici_compl is a dubious translation:

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -161,7 +161,7 @@ theorem isOpen_preimage_ofLower (S : Set α) :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.topologicalSpace.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) => α -> (WithLowerTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => WithLowerTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => WithLowerTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
 Case conversion may be inaccurate. Consider using '#align with_lower_topology.is_open_def WithLowerTopology.isOpen_defₓ'. -/
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
     IsOpen T ↔

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -243,7 +243,12 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 -/
 
-#print LowerTopology.isLowerSet_of_isOpen /-
+/- warning: lower_topology.is_lower_set_of_is_open -> LowerTopology.isLowerSet_of_isOpen is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsOpen.{u1} α _inst_2 s) -> (IsLowerSet.{u1} α (Preorder.toHasLe.{u1} α _inst_1) s)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsOpen.{u1} α _inst_2 s) -> (IsLowerSet.{u1} α (Preorder.toLE.{u1} α _inst_1) s)
+Case conversion may be inaccurate. Consider using '#align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpenₓ'. -/
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   by
@@ -254,13 +259,16 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   case inter u v hu1 hv1 hu2 hv2 => exact hu2.inter hv2
   case sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
--/
 
-#print LowerTopology.isUpperSet_of_isClosed /-
+/- warning: lower_topology.is_upper_set_of_is_closed -> LowerTopology.isUpperSet_of_isClosed is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsClosed.{u1} α _inst_2 s) -> (IsUpperSet.{u1} α (Preorder.toHasLe.{u1} α _inst_1) s)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : LowerTopology.{u1} α _inst_2 _inst_1] {s : Set.{u1} α}, (IsClosed.{u1} α _inst_2 s) -> (IsUpperSet.{u1} α (Preorder.toLE.{u1} α _inst_1) s)
+Case conversion may be inaccurate. Consider using '#align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosedₓ'. -/
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
   isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl
 #align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosed
--/
 
 #print LowerTopology.closure_singleton /-
 /--

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -238,8 +238,8 @@ theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
 /-- The upper closure of a finite set is closed in the lower topology. -/
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) :=
   by
-  simp only [← UpperSet.infᵢ_Ici, UpperSet.coe_infᵢ]
-  exact isClosed_bunionᵢ h fun a h₁ => isClosed_Ici a
+  simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
+  exact isClosed_biUnion h fun a h₁ => isClosed_Ici a
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 -/
 
@@ -252,7 +252,7 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s :=
   case basic u h => obtain ⟨a, rfl⟩ := h; exact (isUpperSet_Ici a).compl
   case univ => exact isLowerSet_univ
   case inter u v hu1 hv1 hu2 hv2 => exact hu2.inter hv2
-  case sUnion _ _ ih => exact isLowerSet_unionₛ ih
+  case sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
 -/
 
@@ -322,7 +322,7 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
     dsimp
     simp_rw [coe_upperClosure, compl_Union, prod_eq, preimage_Inter, preimage_compl]
     -- Note: `refine` doesn't work here because it tries using `prod.topological_space`.
-    apply (isOpen_binterᵢ hs fun a _ => _).inter (isOpen_binterᵢ ht fun b _ => _)
+    apply (isOpen_biInter hs fun a _ => _).inter (isOpen_biInter ht fun b _ => _)
     · exact generate_open.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact generate_open.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
     all_goals infer_instance
@@ -332,22 +332,22 @@ section CompleteLattice
 variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
   [TopologicalSpace β] [LowerTopology β]
 
-/- warning: Inf_hom.continuous -> InfₛHom.continuous is a dubious translation:
+/- warning: Inf_hom.continuous -> sInfHom.continuous is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : CompleteLattice.{u2} β] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : LowerTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_2)))] (f : InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))), Continuous.{u1, u2} α β _inst_3 _inst_5 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) (fun (_x : InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) => α -> β) (InfₛHom.hasCoeToFun.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) f)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : CompleteLattice.{u2} β] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : LowerTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_2)))] (f : sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))), Continuous.{u1, u2} α β _inst_3 _inst_5 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) (fun (_x : sInfHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) => α -> β) (sInfHom.hasCoeToFun.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) f)
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : CompleteLattice.{u1} β] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : LowerTopology.{u2} α _inst_3 (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : LowerTopology.{u1} β _inst_5 (PartialOrder.toPreorder.{u1} β (CompleteSemilatticeInf.toPartialOrder.{u1} β (CompleteLattice.toCompleteSemilatticeInf.{u1} β _inst_2)))] (f : InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))), Continuous.{u2, u1} α β _inst_3 _inst_5 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.374 : α) => β) _x) (InfₛHomClass.toFunLike.{max u2 u1, u2, u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)) (InfₛHom.instInfₛHomClassInfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)))) f)
-Case conversion may be inaccurate. Consider using '#align Inf_hom.continuous InfₛHom.continuousₓ'. -/
-theorem InfₛHom.continuous (f : InfₛHom α β) : Continuous f :=
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : CompleteLattice.{u1} β] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : LowerTopology.{u2} α _inst_3 (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : LowerTopology.{u1} β _inst_5 (PartialOrder.toPreorder.{u1} β (CompleteSemilatticeInf.toPartialOrder.{u1} β (CompleteLattice.toCompleteSemilatticeInf.{u1} β _inst_2)))] (f : sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))), Continuous.{u2, u1} α β _inst_3 _inst_5 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.374 : α) => β) _x) (sInfHomClass.toFunLike.{max u2 u1, u2, u1} (sInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)) (sInfHom.instSInfHomClassSInfHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)))) f)
+Case conversion may be inaccurate. Consider using '#align Inf_hom.continuous sInfHom.continuousₓ'. -/
+theorem sInfHom.continuous (f : sInfHom α β) : Continuous f :=
   by
   convert continuous_generateFrom _
   · exact LowerTopology.topology_eq_lowerTopology β
   rintro _ ⟨b, rfl⟩
   rw [preimage_compl, isOpen_compl_iff]
   convert LowerTopology.isClosed_Ici (Inf <| f ⁻¹' Ici b)
-  refine' subset_antisymm (fun a => infₛ_le) fun a ha => le_trans _ <| OrderHomClass.mono f ha
+  refine' subset_antisymm (fun a => sInf_le) fun a ha => le_trans _ <| OrderHomClass.mono f ha
   simp [map_Inf]
-#align Inf_hom.continuous InfₛHom.continuous
+#align Inf_hom.continuous sInfHom.continuous
 
 /- warning: lower_topology.to_has_continuous_inf -> LowerTopology.continuousInf is a dubious translation:
 lean 3 declaration is
@@ -357,7 +357,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align lower_topology.to_has_continuous_inf LowerTopology.continuousInfₓ'. -/
 -- see Note [lower instance priority]
 instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
-  ⟨(infInfₛHom : InfₛHom (α × α) α).Continuous⟩
+  ⟨(infsInfHom : sInfHom (α × α) α).Continuous⟩
 #align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
 
 end CompleteLattice

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -172,7 +172,7 @@ theorem isOpen_def (T : Set (WithLowerTopology α)) :
 end WithLowerTopology
 
 #print LowerTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`topology_eq_lowerTopology] [] -/
 /--
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -213,7 +213,7 @@ def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
       convert continuous_id
       apply topology_eq_lower_topology
     continuous_invFun := by
-      convert ← continuous_id
+      convert← continuous_id
       apply topology_eq_lower_topology }
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -161,7 +161,7 @@ theorem isOpen_preimage_ofLower (S : Set α) :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.topologicalSpace.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) => α -> (WithLowerTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : α) => WithLowerTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
+  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] (T : Set.{u1} (WithLowerTopology.{u1} α)), Iff (IsOpen.{u1} (WithLowerTopology.{u1} α) (WithLowerTopology.instTopologicalSpaceWithLowerTopology.{u1} α _inst_1) T) (TopologicalSpace.IsOpen.{u1} α (TopologicalSpace.generateFrom.{u1} α (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{succ u1} α (fun (a : α) => Eq.{succ u1} (Set.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Set.Ici.{u1} α _inst_1 a)) s)))) (Set.preimage.{u1, u1} α (WithLowerTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => WithLowerTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (WithLowerTopology.{u1} α)) (WithLowerTopology.toLower.{u1} α)) T))
 Case conversion may be inaccurate. Consider using '#align with_lower_topology.is_open_def WithLowerTopology.isOpen_defₓ'. -/
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
     IsOpen T ↔
@@ -336,7 +336,7 @@ variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerT
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : CompleteLattice.{u2} β] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : LowerTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_2)))] (f : InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))), Continuous.{u1, u2} α β _inst_3 _inst_5 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) (fun (_x : InfₛHom.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) => α -> β) (InfₛHom.hasCoeToFun.{u1, u2} α β (ConditionallyCompleteLattice.toHasInf.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (ConditionallyCompleteLattice.toHasInf.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_2))) f)
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : CompleteLattice.{u1} β] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : LowerTopology.{u2} α _inst_3 (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : LowerTopology.{u1} β _inst_5 (PartialOrder.toPreorder.{u1} β (CompleteSemilatticeInf.toPartialOrder.{u1} β (CompleteLattice.toCompleteSemilatticeInf.{u1} β _inst_2)))] (f : InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))), Continuous.{u2, u1} α β _inst_3 _inst_5 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.372 : α) => β) _x) (InfₛHomClass.toFunLike.{max u2 u1, u2, u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)) (InfₛHom.instInfₛHomClassInfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)))) f)
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : CompleteLattice.{u2} α] [_inst_2 : CompleteLattice.{u1} β] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : LowerTopology.{u2} α _inst_3 (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : LowerTopology.{u1} β _inst_5 (PartialOrder.toPreorder.{u1} β (CompleteSemilatticeInf.toPartialOrder.{u1} β (CompleteLattice.toCompleteSemilatticeInf.{u1} β _inst_2)))] (f : InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))), Continuous.{u2, u1} α β _inst_3 _inst_5 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.374 : α) => β) _x) (InfₛHomClass.toFunLike.{max u2 u1, u2, u1} (InfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2))) α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)) (InfₛHom.instInfₛHomClassInfₛHom.{u2, u1} α β (ConditionallyCompleteLattice.toInfSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (ConditionallyCompleteLattice.toInfSet.{u1} β (CompleteLattice.toConditionallyCompleteLattice.{u1} β _inst_2)))) f)
 Case conversion may be inaccurate. Consider using '#align Inf_hom.continuous InfₛHom.continuousₓ'. -/
 theorem InfₛHom.continuous (f : InfₛHom α β) : Continuous f :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -349,12 +349,16 @@ theorem InfₛHom.continuous (f : InfₛHom α β) : Continuous f :=
   simp [map_Inf]
 #align Inf_hom.continuous InfₛHom.continuous
 
-#print LowerTopology.continuousInf /-
+/- warning: lower_topology.to_has_continuous_inf -> LowerTopology.continuousInf is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))], ContinuousInf.{u1} α _inst_3 (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : LowerTopology.{u1} α _inst_3 (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))], ContinuousInf.{u1} α _inst_3 (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align lower_topology.to_has_continuous_inf LowerTopology.continuousInfₓ'. -/
 -- see Note [lower instance priority]
 instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
   ⟨(infInfₛHom : InfₛHom (α × α) α).Continuous⟩
 #align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
--/
 
 end CompleteLattice
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

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".

@@ -98,7 +98,7 @@ namespace WithLower
 lemma toLower_inj {a b : α} : toLower a = toLower b ↔ a = b := Iff.rfl
 #align with_lower_topology.to_lower_inj Topology.WithLower.toLower_inj
 
--- porting note: removed @[simp] to make linter happy
+-- Porting note: removed @[simp] to make linter happy
 theorem ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b :=
   Iff.rfl
 #align with_lower_topology.of_lower_inj Topology.WithLower.ofLower_inj
@@ -239,7 +239,7 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
-  -- porting note: `rw` leaves a shadowed assumption
+  -- Porting note: `rw` leaves a shadowed assumption
   replace h := isOpen_iff_generate_Ici_compl.1 h
   induction h with
   | basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl

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>

@@ -412,8 +412,7 @@ instance instIsUpperProd [Preorder α] [TopologicalSpace α] [IsUpper α]
     [OrderTop α] [Preorder β] [TopologicalSpace β] [IsUpper β] [OrderTop β] :
     IsUpper (α × β) where
   topology_eq_upperTopology := by
-    suffices : IsLower (α × β)ᵒᵈ
-    · exact IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
+    suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
     exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ)
 
 section CompleteLattice_IsLower

• Make type explicit in def of Scott-Hausdorff and Scott topologies
• Move With{Lower|Upper}Homeomorph into the correct namespace

Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com>

@@ -211,9 +211,9 @@ variable {α}
 
 /-- If `α` is equipped with the lower topology, then it is homeomorphic to `WithLower α`.
 -/
-def WithLowerHomeomorph : WithLower α ≃ₜ α :=
+def withLowerHomeomorph : WithLower α ≃ₜ α :=
   WithLower.ofLower.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
-#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.WithLowerHomeomorph
+#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.withLowerHomeomorph
 
 theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s := by
   rw [topology_eq α]; rfl
@@ -324,7 +324,7 @@ variable {α}
 
 /-- If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpper α`.
 -/
-def WithUpperHomeomorph : WithUpper α ≃ₜ α :=
+def withUpperHomeomorph : WithUpper α ≃ₜ α :=
   WithUpper.ofUpper.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
 
 theorem isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s := by

This converts usages of the pattern

cases h
case inl h' => ...
case inr h' => ...

which derive from mathported code, to the "structured cases" syntax:

cases h with
| inl h' => ...
| inr h' => ...

The case where the subgoals are handled with · instead of case is more contentious (and much more numerous) so I left those alone. This pattern also appears with cases', induction, induction', and rcases. Furthermore, there is a similar transformation for by_cases:

by_cases h : cond
case pos => ...
case neg => ...

is replaced by:

if h : cond then
  ...
else
  ...

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -241,11 +241,11 @@ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
   -- porting note: `rw` leaves a shadowed assumption
   replace h := isOpen_iff_generate_Ici_compl.1 h
-  induction h
-  case basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl
-  case univ => exact isLowerSet_univ
-  case inter u v _ _ hu2 hv2 => exact hu2.inter hv2
-  case sUnion _ _ ih => exact isLowerSet_sUnion ih
+  induction h with
+  | basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl
+  | univ => exact isLowerSet_univ
+  | inter u v _ _ hu2 hv2 => exact hu2.inter hv2
+  | sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open Topology.IsLower.isLowerSet_of_isOpen
 
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=

Similarly, upgrade tendsto_nhds_generateFrom, IsTopologicalBasis.continuous, Topology.IsLower.continuous_of_Ici, and Topology.IsUpper.continuous_iff_Iic.

The old lemmas are now deprecated, and the new ones have _iff in their names. Once we remove the old lemmas, we can drop the _iff suffixes.

@@ -277,16 +277,17 @@ protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
     exact ⟨_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]⟩
 #align lower_topology.is_topological_basis Topology.IsLower.isTopologicalBasis
 
-/-- A function `f : β → α` with lower topology in the codomain is continuous provided that the
-preimage of every interval `Set.Ici a` is a closed set.
-
-TODO: upgrade to an `iff`. -/
-lemma continuous_of_Ici [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Ici a))) :
-    Continuous f := by
+/-- A function `f : β → α` with lower topology in the codomain is continuous
+if and only if the preimage of every interval `Set.Ici a` is a closed set.
+-/
+lemma continuous_iff_Ici [TopologicalSpace β] {f : β → α} :
+    Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Ici a)) := by
   obtain rfl := IsLower.topology_eq α
-  refine continuous_generateFrom ?_
-  rintro _ ⟨a, rfl⟩
-  exact (h a).isOpen_compl
+  simp [continuous_generateFrom_iff]
+
+/-- A function `f : β → α` with lower topology in the codomain is continuous provided that the
+preimage of every interval `Set.Ici a` is a closed set. -/
+@[deprecated] alias ⟨_, continuous_of_Ici⟩ := continuous_iff_Ici
 
 end Preorder
 
@@ -359,13 +360,18 @@ theorem closure_singleton (a : α) : closure {a} = Iic a :=
 protected theorem isTopologicalBasis : IsTopologicalBasis (upperBasis α) :=
   IsLower.isTopologicalBasis (α := αᵒᵈ)
 
-/-- A function `f : β → α` with upper topology in the codomain is continuous provided that the
-preimage of every interval `Set.Iic a` is a closed set.
+/-- A function `f : β → α` with upper topology in the codomain is continuous
+if and only if the preimage of every interval `Set.Iic a` is a closed set. -/
+lemma continuous_iff_Iic [TopologicalSpace β] {f : β → α} :
+    Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) :=
+  IsLower.continuous_iff_Ici (α := αᵒᵈ)
 
-TODO: upgrade to an `iff`. -/
+/-- A function `f : β → α` with upper topology in the codomain is continuous
+provided that the preimage of every interval `Set.Iic a` is a closed set. -/
+@[deprecated]
 lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :
     Continuous f :=
-  IsLower.continuous_of_Ici (α := αᵒᵈ) h
+  continuous_iff_Iic.2 h
 
 end Preorder
 
@@ -416,7 +422,7 @@ variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [IsLowe
   [TopologicalSpace β] [IsLower β]
 
 protected lemma _root_.sInfHom.continuous (f : sInfHom α β) : Continuous f := by
-  refine IsLower.continuous_of_Ici fun b => ?_
+  refine IsLower.continuous_iff_Ici.2 fun b => ?_
   convert isClosed_Ici (a := sInf <| f ⁻¹' Ici b)
   refine' Subset.antisymm (fun a => sInf_le) fun a ha => le_trans _ <|
     OrderHomClass.mono (f : α →o β) ha

Rename the Upper/Lower UpperSet/LowerSet topologies to the conventions suggested by @YaelDillies in https://github.com/leanprover-community/mathlib4/pull/2508#discussion_r1306711450

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -3,9 +3,9 @@ Copyright (c) 2023 Christopher Hoskin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
 -/
+import Mathlib.Order.Hom.CompleteLattice
 import Mathlib.Topology.Homeomorph
 import Mathlib.Topology.Order.Lattice
-import Mathlib.Order.Hom.CompleteLattice
 
 #align_import topology.order.lower_topology from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
 
@@ -20,20 +20,20 @@ right-closed left-infinite intervals.
 
 ## Main statements
 
-- `LowerTopology.t0Space` - the lower topology on a partial order is T₀
-- `LowerTopology.isTopologicalBasis` - the complements of the upper closures of finite
+- `IsLower.t0Space` - the lower topology on a partial order is T₀
+- `IsLower.isTopologicalBasis` - the complements of the upper closures of finite
   subsets form a basis for the lower topology
-- `LowerTopology.continuousInf` - the inf map is continuous with respect to the lower topology
+- `IsLower.continuousInf` - the inf map is continuous with respect to the lower topology
 
 ## Implementation notes
 
-A type synonym `WithLowerTopology` is introduced and for a preorder `α`, `WithLowerTopology α`
+A type synonym `WithLower` is introduced and for a preorder `α`, `WithLower α`
 is made an instance of `TopologicalSpace` by the topology generated by the complements of the
 closed intervals to infinity.
 
-We define a mixin class `LowerTopology` for the class of types which are both a preorder and a
+We define a mixin class `IsLower` for the class of types which are both a preorder and a
 topology and where the topology is generated by the complements of the closed intervals to infinity.
-It is shown that `WithLowerTopology α` is an instance of `LowerTopology`.
+It is shown that `WithLower α` is an instance of `IsLower`.
 
 Similarly for the upper topology.
 
@@ -51,196 +51,140 @@ of the lower topology to the spectrum of a complete lattice coincides with the h
 lower topology, upper topology, preorder
 -/
 
-set_option autoImplicit true
+open Set TopologicalSpace Topology
 
-open Set TopologicalSpace
+namespace Topology
 
-section WithLowerTopology
+/--
+The lower topology is the topology generated by the complements of the left-closed right-infinite
+intervals.
+-/
+def lower (α : Type*) [Preorder α] : TopologicalSpace α := generateFrom {s | ∃ a, (Ici a)ᶜ = s}
 
-variable (α β : Type*)
+/--
+The upper topology is the topology generated by the complements of the right-closed left-infinite
+intervals.
+-/
+def upper (α : Type*) [Preorder α] : TopologicalSpace α := generateFrom {s | ∃ a, (Iic a)ᶜ = s}
 
-/-- Type synonym for a preorder equipped with the lower topology. -/
-def WithLowerTopology := α
-#align with_lower_topology WithLowerTopology
+/-- Type synonym for a preorder equipped with the lower set topology. -/
+def WithLower (α : Type*) := α
+#align with_lower_topology Topology.WithLower
 
 variable {α β}
 
-namespace WithLowerTopology
+namespace WithLower
 
-/-- `toLower` is the identity function to the `WithLowerTopology` of a type.  -/
-@[match_pattern]
-def toLower : α ≃ WithLowerTopology α := Equiv.refl _
-#align with_lower_topology.to_lower WithLowerTopology.toLower
+/-- `toLower` is the identity function to the `WithLower` of a type.  -/
+@[match_pattern] def toLower : α ≃ WithLower α := Equiv.refl _
+#align with_lower_topology.to_lower Topology.WithLower.toLower
 
-/-- `ofLower` is the identity function from the `WithLowerTopology` of a type.  -/
-@[match_pattern]
-def ofLower : WithLowerTopology α ≃ α := Equiv.refl _
-#align with_lower_topology.of_lower WithLowerTopology.ofLower
+/-- `ofLower` is the identity function from the `WithLower` of a type.  -/
+@[match_pattern] def ofLower : WithLower α ≃ α := Equiv.refl _
+#align with_lower_topology.of_lower Topology.WithLower.ofLower
 
-@[simp]
-theorem to_withLowerTopology_symm_eq : (@toLower α).symm = ofLower :=
-  rfl
-#align with_lower_topology.to_with_lower_topology_symm_eq WithLowerTopology.to_withLowerTopology_symm_eq
+@[simp] lemma to_WithLower_symm_eq : (@toLower α).symm = ofLower := rfl
+#align with_lower_topology.to_with_lower_topology_symm_eq Topology.WithLower.to_WithLower_symm_eq
 
-@[simp]
-theorem of_withLowerTopology_symm_eq : (@ofLower α).symm = toLower :=
-  rfl
-#align with_lower_topology.of_with_lower_topology_symm_eq WithLowerTopology.of_withLowerTopology_symm_eq
+@[simp] lemma of_WithLower_symm_eq : (@ofLower α).symm = toLower := rfl
+#align with_lower_topology.of_with_lower_topology_symm_eq Topology.WithLower.of_WithLower_symm_eq
 
-@[simp]
-theorem toLower_ofLower (a : WithLowerTopology α) : toLower (ofLower a) = a :=
-  rfl
-#align with_lower_topology.to_lower_of_lower WithLowerTopology.toLower_ofLower
+@[simp] lemma toLower_ofLower (a : WithLower α) : toLower (ofLower a) = a := rfl
+#align with_lower_topology.to_lower_of_lower Topology.WithLower.toLower_ofLower
 
-@[simp]
-theorem ofLower_toLower (a : α) : ofLower (toLower a) = a :=
-  rfl
-#align with_lower_topology.of_lower_to_lower WithLowerTopology.ofLower_toLower
+@[simp] lemma ofLower_toLower (a : α) : ofLower (toLower a) = a := rfl
+#align with_lower_topology.of_lower_to_lower Topology.WithLower.ofLower_toLower
 
--- porting note: removed @[simp] to make linter happy
-theorem toLower_inj {a b : α} : toLower a = toLower b ↔ a = b :=
-  Iff.rfl
-#align with_lower_topology.to_lower_inj WithLowerTopology.toLower_inj
+lemma toLower_inj {a b : α} : toLower a = toLower b ↔ a = b := Iff.rfl
+#align with_lower_topology.to_lower_inj Topology.WithLower.toLower_inj
 
 -- porting note: removed @[simp] to make linter happy
-theorem ofLower_inj {a b : WithLowerTopology α} : ofLower a = ofLower b ↔ a = b :=
+theorem ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b :=
   Iff.rfl
-#align with_lower_topology.of_lower_inj WithLowerTopology.ofLower_inj
+#align with_lower_topology.of_lower_inj Topology.WithLower.ofLower_inj
 
-/-- A recursor for `WithLowerTopology`. Use as `induction x using WithLowerTopology.rec`. -/
-protected def rec {β : WithLowerTopology α → Sort*} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
+/-- A recursor for `WithLower`. Use as `induction x using WithLower.rec`. -/
+protected def rec {β : WithLower α → Sort*} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
   h (ofLower a)
-#align with_lower_topology.rec WithLowerTopology.rec
-
-instance [Nonempty α] : Nonempty (WithLowerTopology α) :=
-  ‹Nonempty α›
+#align with_lower_topology.rec Topology.WithLower.rec
 
-instance [Inhabited α] : Inhabited (WithLowerTopology α) :=
-  ‹Inhabited α›
+instance [Nonempty α] : Nonempty (WithLower α) := ‹Nonempty α›
+instance [Inhabited α] : Inhabited (WithLower α) := ‹Inhabited α›
 
-variable [Preorder α]
+variable [Preorder α] {s : Set α}
 
-instance : Preorder (WithLowerTopology α) :=
-  ‹Preorder α›
+instance : Preorder (WithLower α) := ‹Preorder α›
+instance : TopologicalSpace (WithLower α) := lower α
 
-instance : TopologicalSpace (WithLowerTopology α) :=
-  generateFrom { s | ∃ a, (Ici a)ᶜ = s }
+lemma isOpen_preimage_ofLower : IsOpen (ofLower ⁻¹' s) ↔ (lower α).IsOpen s := Iff.rfl
+#align with_lower_topology.is_open_preimage_of_lower Topology.WithLower.isOpen_preimage_ofLower
 
-theorem isOpen_preimage_ofLower (S : Set α) :
-    IsOpen (WithLowerTopology.ofLower ⁻¹' S) ↔
-      (generateFrom { s : Set α | ∃ a : α, (Ici a)ᶜ = s }).IsOpen S :=
+lemma isOpen_def (T : Set (WithLower α)) : IsOpen T ↔ (lower α).IsOpen (WithLower.toLower ⁻¹' T) :=
   Iff.rfl
-#align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
+#align with_lower_topology.is_open_def Topology.WithLower.isOpen_def
 
-theorem isOpen_def (T : Set (WithLowerTopology α)) :
-    IsOpen T ↔ (generateFrom { s : Set α | ∃ a : α, (Ici a)ᶜ = s }).IsOpen
-    (WithLowerTopology.toLower ⁻¹' T) :=
-  Iff.rfl
-#align with_lower_topology.is_open_def WithLowerTopology.isOpen_def
-
-end WithLowerTopology
-
-end WithLowerTopology
-
-section WithUpperTopology
-
-variable (α β : Type*)
+end WithLower
 
 /-- Type synonym for a preorder equipped with the upper topology. -/
-def WithUpperTopology := α
+def WithUpper (α : Type*) := α
+namespace WithUpper
 
-variable {α β}
+/-- `toUpper` is the identity function to the `WithUpper` of a type.  -/
+@[match_pattern] def toUpper : α ≃ WithUpper α := Equiv.refl _
 
-namespace WithUpperTopology
+/-- `ofUpper` is the identity function from the `WithUpper` of a type.  -/
+@[match_pattern] def ofUpper : WithUpper α ≃ α := Equiv.refl _
 
-/-- `toUpper` is the identity function to the `WithUpperTopology` of a type.  -/
-@[match_pattern]
-def toUpper : α ≃ WithUpperTopology α := Equiv.refl _
+@[simp] lemma to_WithUpper_symm_eq {α} : (@toUpper α).symm = ofUpper := rfl
+@[simp] lemma of_WithUpper_symm_eq : (@ofUpper α).symm = toUpper := rfl
+@[simp] lemma toUpper_ofUpper (a : WithUpper α) : toUpper (ofUpper a) = a := rfl
+@[simp] lemma ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a := rfl
+lemma toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b := Iff.rfl
+lemma ofUpper_inj {a b : WithUpper α} : ofUpper a = ofUpper b ↔ a = b := Iff.rfl
 
-/-- `ofUpper` is the identity function from the `WithUpperTopology` of a type.  -/
-@[match_pattern]
-def ofUpper : WithUpperTopology α ≃ α := Equiv.refl _
-
-@[simp]
-theorem to_withUpperTopology_symm_eq {α} : (@toUpper α).symm = ofUpper :=
-  rfl
-
-@[simp]
-theorem of_withUpperTopology_symm_eq : (@ofUpper α).symm = toUpper :=
-  rfl
-
-@[simp]
-theorem toUpper_ofUpper (a : WithUpperTopology α) : toUpper (ofUpper a) = a :=
-  rfl
-
-@[simp]
-theorem ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a :=
-  rfl
-
-theorem toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b :=
-  Iff.rfl
-
-theorem ofUpper_inj {a b : WithUpperTopology α} : ofUpper a = ofUpper b ↔ a = b :=
-  Iff.rfl
-
-/-- A recursor for `WithUpperTopology`. Use as `induction x using WithUpperTopology.rec`. -/
-protected def rec {β : WithUpperTopology α → Sort*} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
+/-- A recursor for `WithUpper`. Use as `induction x using WithUpper.rec`. -/
+protected def rec {β : WithUpper α → Sort*} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
   h (ofUpper a)
 
-instance [Nonempty α] : Nonempty (WithUpperTopology α) :=
-  ‹Nonempty α›
-
-instance [Inhabited α] : Inhabited (WithUpperTopology α) :=
-  ‹Inhabited α›
+instance [Nonempty α] : Nonempty (WithUpper α) := ‹Nonempty α›
+instance [Inhabited α] : Inhabited (WithUpper α) := ‹Inhabited α›
 
-variable [Preorder α]
+variable [Preorder α] {s : Set α}
 
-instance : Preorder (WithUpperTopology α) :=
-  ‹Preorder α›
+instance : Preorder (WithUpper α) := ‹Preorder α›
+instance : TopologicalSpace (WithUpper α) := upper α
 
-instance : TopologicalSpace (WithUpperTopology α) :=
-  generateFrom { s | ∃ a, (Iic a)ᶜ = s }
+lemma isOpen_preimage_ofUpper : IsOpen (ofUpper ⁻¹' s) ↔ (upper α).IsOpen s := Iff.rfl
 
-theorem isOpen_preimage_ofUpper (S : Set α) :
-    IsOpen (WithUpperTopology.ofUpper ⁻¹' S) ↔
-      (generateFrom { s : Set α | ∃ a : α, (Iic a)ᶜ = s }).IsOpen S :=
-  Iff.rfl
-
-theorem isOpen_def (T : Set (WithUpperTopology α)) :
-    IsOpen T ↔ (generateFrom { s : Set α | ∃ a : α, (Iic a)ᶜ = s }).IsOpen
-    (WithUpperTopology.toUpper ⁻¹' T) :=
-  Iff.rfl
+lemma isOpen_def {s : Set (WithUpper α)} : IsOpen s ↔ (upper α).IsOpen (toUpper ⁻¹' s) := Iff.rfl
 
-end WithUpperTopology
-
-end WithUpperTopology
+end WithUpper
 
 /--
 The lower topology is the topology generated by the complements of the left-closed right-infinite
 intervals.
 -/
-class LowerTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
-  topology_eq_lowerTopology : t = generateFrom { s | ∃ a, (Ici a)ᶜ = s }
-#align lower_topology LowerTopology
+class IsLower (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
+  topology_eq_lowerTopology : t = lower α
+#align lower_topology Topology.IsLower
+
+attribute [nolint docBlame] IsLower.topology_eq_lowerTopology
 
 /--
 The upper topology is the topology generated by the complements of the right-closed left-infinite
 intervals.
 -/
-class UpperTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
-  topology_eq_upperTopology : t = generateFrom { s | ∃ a, (Iic a)ᶜ = s }
-
-instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
-  ⟨rfl⟩
+class IsUpper (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
+  topology_eq_upperTopology : t = upper α
+attribute [nolint docBlame] IsUpper.topology_eq_upperTopology
 
-instance [Preorder α] : UpperTopology (WithUpperTopology α) :=
-  ⟨rfl⟩
+instance [Preorder α] : IsLower (WithLower α) := ⟨rfl⟩
+instance [Preorder α] : IsUpper (WithUpper α) := ⟨rfl⟩
 
 /--
 The lower topology is homeomorphic to the upper topology on the dual order
 -/
-def WithLowerTopology.toDualHomeomorph [Preorder α] : WithLowerTopology α ≃ₜ WithUpperTopology αᵒᵈ
-    where
+def WithLower.toDualHomeomorph [Preorder α] : WithLower α ≃ₜ WithUpper αᵒᵈ where
   toFun := OrderDual.toDual
   invFun := OrderDual.ofDual
   left_inv := OrderDual.toDual_ofDual
@@ -248,42 +192,42 @@ def WithLowerTopology.toDualHomeomorph [Preorder α] : WithLowerTopology α ≃
   continuous_toFun := continuous_coinduced_rng
   continuous_invFun := continuous_coinduced_rng
 
-namespace LowerTopology
+namespace IsLower
 
 /-- The complements of the upper closures of finite sets are a collection of lower sets
 which form a basis for the lower topology. -/
 def lowerBasis (α : Type*) [Preorder α] :=
   { s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s }
-#align lower_topology.lower_basis LowerTopology.lowerBasis
+#align lower_topology.lower_basis Topology.IsLower.lowerBasis
 
 section Preorder
 
 variable (α)
-variable [Preorder α] [TopologicalSpace α] [LowerTopology α] {s : Set α}
+variable [Preorder α] [TopologicalSpace α] [IsLower α] {s : Set α}
 
-lemma topology_eq : ‹_› = generateFrom { s | ∃ a : α, (Ici a)ᶜ = s } := topology_eq_lowerTopology
+lemma topology_eq : ‹_› = lower α := topology_eq_lowerTopology
 
 variable {α}
 
-/-- If `α` is equipped with the lower topology, then it is homeomorphic to `WithLowerTopology α`.
+/-- If `α` is equipped with the lower topology, then it is homeomorphic to `WithLower α`.
 -/
-def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
-  WithLowerTopology.ofLower.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
-#align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
+def WithLowerHomeomorph : WithLower α ≃ₜ α :=
+  WithLower.ofLower.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
+#align lower_topology.with_lower_topology_homeomorph Topology.IsLower.WithLowerHomeomorph
 
 theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s := by
   rw [topology_eq α]; rfl
-#align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
+#align lower_topology.is_open_iff_generate_Ici_compl Topology.IsLower.isOpen_iff_generate_Ici_compl
 
-instance instUpperTopologyDual [Preorder α] [TopologicalSpace α] [LowerTopology α] :
-    UpperTopology (αᵒᵈ) where
+instance _root_.OrderDual.instIsUpper [Preorder α] [TopologicalSpace α] [IsLower α] :
+    IsUpper αᵒᵈ where
   topology_eq_upperTopology := topology_eq_lowerTopology (α := α)
 
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
 instance : ClosedIciTopology α :=
   ⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩
 
--- Porting note: The old `LowerTopology.isClosed_Ici` was removed, since one can now use
+-- Porting note: The old `IsLower.isClosed_Ici` was removed, since one can now use
 -- the general `isClosed_Ici` lemma thanks to the instance above.
 #align lower_topology.is_closed_Ici isClosed_Ici
 
@@ -291,7 +235,7 @@ instance : ClosedIciTopology α :=
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by
   simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
   exact h.isClosed_biUnion fun _ _ => isClosed_Ici
-#align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
+#align lower_topology.is_closed_upper_closure Topology.IsLower.isClosed_upperClosure
 
 /-- Every set open in the lower topology is a lower set. -/
 theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
@@ -302,11 +246,11 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
   case univ => exact isLowerSet_univ
   case inter u v _ _ hu2 hv2 => exact hu2.inter hv2
   case sUnion _ _ ih => exact isLowerSet_sUnion ih
-#align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
+#align lower_topology.is_lower_set_of_is_open Topology.IsLower.isLowerSet_of_isOpen
 
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
   isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl
-#align lower_topology.is_upper_set_of_is_closed LowerTopology.isUpperSet_of_isClosed
+#align lower_topology.is_upper_set_of_is_closed Topology.IsLower.isUpperSet_of_isClosed
 
 /--
 The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval
@@ -316,7 +260,7 @@ The closure of a singleton `{a}` in the lower topology is the left-closed right-
 theorem closure_singleton (a : α) : closure {a} = Ici a :=
   Subset.antisymm ((closure_minimal fun _ h => h.ge) <| isClosed_Ici) <|
     (isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl
-#align lower_topology.closure_singleton LowerTopology.closure_singleton
+#align lower_topology.closure_singleton Topology.IsLower.closure_singleton
 
 protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
   convert isTopologicalBasis_of_subbasis (topology_eq α)
@@ -331,7 +275,7 @@ protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
     rw [subset_def, Subtype.forall'] at hs
     choose f hf using hs
     exact ⟨_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]⟩
-#align lower_topology.is_topological_basis LowerTopology.isTopologicalBasis
+#align lower_topology.is_topological_basis Topology.IsLower.isTopologicalBasis
 
 /-- A function `f : β → α` with lower topology in the codomain is continuous provided that the
 preimage of every interval `Set.Ici a` is a closed set.
@@ -339,7 +283,7 @@ preimage of every interval `Set.Ici a` is a closed set.
 TODO: upgrade to an `iff`. -/
 lemma continuous_of_Ici [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Ici a))) :
     Continuous f := by
-  obtain rfl := LowerTopology.topology_eq α
+  obtain rfl := IsLower.topology_eq α
   refine continuous_generateFrom ?_
   rintro _ ⟨a, rfl⟩
   exact (h a).isOpen_compl
@@ -348,7 +292,7 @@ end Preorder
 
 section PartialOrder
 
-variable [PartialOrder α] [TopologicalSpace α] [LowerTopology α]
+variable [PartialOrder α] [TopologicalSpace α] [IsLower α]
 
 -- see Note [lower instance priority]
 /-- The lower topology on a partial order is T₀. -/
@@ -358,10 +302,10 @@ instance (priority := 90) t0Space : T0Space α :=
 
 end PartialOrder
 
-end LowerTopology
+end IsLower
 
 
-namespace UpperTopology
+namespace IsUpper
 
 /-- The complements of the lower closures of finite sets are a collection of upper sets
 which form a basis for the upper topology. -/
@@ -371,22 +315,22 @@ def upperBasis (α : Type*) [Preorder α] :=
 section Preorder
 
 variable (α)
-variable [Preorder α] [TopologicalSpace α] [UpperTopology α] {s : Set α}
+variable [Preorder α] [TopologicalSpace α] [IsUpper α] {s : Set α}
 
-lemma topology_eq : ‹_› = generateFrom { s | ∃ a : α, (Iic a)ᶜ = s } := topology_eq_upperTopology
+lemma topology_eq : ‹_› = upper α := topology_eq_upperTopology
 
 variable {α}
 
-/-- If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpperTopology α`.
+/-- If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpper α`.
 -/
-def withUpperTopologyHomeomorph : WithUpperTopology α ≃ₜ α :=
-  WithUpperTopology.ofUpper.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
+def WithUpperHomeomorph : WithUpper α ≃ₜ α :=
+  WithUpper.ofUpper.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
 
 theorem isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s := by
   rw [topology_eq α]; rfl
 
-instance instLowerTopologyDual [Preorder α] [TopologicalSpace α] [UpperTopology α] :
-    LowerTopology (αᵒᵈ) where
+instance _root_.OrderDual.instIsLower [Preorder α] [TopologicalSpace α] [IsUpper α] :
+    IsLower αᵒᵈ where
   topology_eq_lowerTopology := topology_eq_upperTopology (α := α)
 
 /-- Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/
@@ -395,11 +339,11 @@ instance : ClosedIicTopology α :=
 
 /-- The lower closure of a finite set is closed in the upper topology. -/
 theorem isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) :=
-  LowerTopology.isClosed_upperClosure (α := αᵒᵈ) h
+  IsLower.isClosed_upperClosure (α := αᵒᵈ) h
 
 /-- Every set open in the upper topology is a upper set. -/
 theorem isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s :=
-  LowerTopology.isLowerSet_of_isOpen (α := αᵒᵈ) h
+  IsLower.isLowerSet_of_isOpen (α := αᵒᵈ) h
 
 theorem isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s :=
   isUpperSet_compl.1 <| isUpperSet_of_isOpen h.isOpen_compl
@@ -410,10 +354,10 @@ The closure of a singleton `{a}` in the upper topology is the left-infinite righ
 -/
 @[simp]
 theorem closure_singleton (a : α) : closure {a} = Iic a :=
-  LowerTopology.closure_singleton (α := αᵒᵈ) _
+  IsLower.closure_singleton (α := αᵒᵈ) _
 
 protected theorem isTopologicalBasis : IsTopologicalBasis (upperBasis α) :=
-  LowerTopology.isTopologicalBasis (α := αᵒᵈ)
+  IsLower.isTopologicalBasis (α := αᵒᵈ)
 
 /-- A function `f : β → α` with upper topology in the codomain is continuous provided that the
 preimage of every interval `Set.Iic a` is a closed set.
@@ -421,55 +365,58 @@ preimage of every interval `Set.Iic a` is a closed set.
 TODO: upgrade to an `iff`. -/
 lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :
     Continuous f :=
-  LowerTopology.continuous_of_Ici (α := αᵒᵈ) h
+  IsLower.continuous_of_Ici (α := αᵒᵈ) h
 
 end Preorder
 
 
 section PartialOrder
 
-variable [PartialOrder α] [TopologicalSpace α] [UpperTopology α]
+variable [PartialOrder α] [TopologicalSpace α] [IsUpper α]
 
 -- see Note [lower instance priority]
 /-- The upper topology on a partial order is T₀. -/
 instance (priority := 90) t0Space : T0Space α :=
-  LowerTopology.t0Space (α := αᵒᵈ)
+  IsLower.t0Space (α := αᵒᵈ)
 
 end PartialOrder
 
-end UpperTopology
+end IsUpper
 
-instance instLowerTopologyProd [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α]
-    [Preorder β] [TopologicalSpace β] [LowerTopology β] [OrderBot β] : LowerTopology (α × β) where
+instance instIsLowerProd [Preorder α] [TopologicalSpace α] [IsLower α]
+    [OrderBot α] [Preorder β] [TopologicalSpace β] [IsLower β] [OrderBot β] :
+    IsLower (α × β) where
   topology_eq_lowerTopology := by
     refine' le_antisymm (le_generateFrom _) _
     · rintro _ ⟨x, rfl⟩
       exact (isClosed_Ici.prod isClosed_Ici).isOpen_compl
-    rw [(LowerTopology.isTopologicalBasis.prod LowerTopology.isTopologicalBasis).eq_generateFrom,
-      le_generateFrom_iff_subset_isOpen, image2_subset_iff]
+    rw [(IsLower.isTopologicalBasis.prod
+      IsLower.isTopologicalBasis).eq_generateFrom, le_generateFrom_iff_subset_isOpen,
+      image2_subset_iff]
     rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
     dsimp
     simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]
     -- without `let`, `refine` tries to use the product topology and fails
-    let _ : TopologicalSpace (α × β) := generateFrom { s | ∃ a, (Ici a)ᶜ = s }
+    let _ : TopologicalSpace (α × β) := lower (α × β)
     refine (hs.isOpen_biInter fun a _ => ?_).inter (ht.isOpen_biInter fun b _ => ?_)
     · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
 
-instance instUpperTopologyProd [Preorder α] [TopologicalSpace α] [UpperTopology α] [OrderTop α]
-    [Preorder β] [TopologicalSpace β] [UpperTopology β] [OrderTop β] : UpperTopology (α × β) where
+instance instIsUpperProd [Preorder α] [TopologicalSpace α] [IsUpper α]
+    [OrderTop α] [Preorder β] [TopologicalSpace β] [IsUpper β] [OrderTop β] :
+    IsUpper (α × β) where
   topology_eq_upperTopology := by
-    suffices : LowerTopology (α × β)ᵒᵈ
-    · exact LowerTopology.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
-    exact instLowerTopologyProd (α := αᵒᵈ) (β := βᵒᵈ)
+    suffices : IsLower (α × β)ᵒᵈ
+    · exact IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
+    exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ)
 
-section CompleteLattice_LowerTopology
+section CompleteLattice_IsLower
 
-variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
-  [TopologicalSpace β] [LowerTopology β]
+variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [IsLower α]
+  [TopologicalSpace β] [IsLower β]
 
-protected theorem sInfHom.continuous (f : sInfHom α β) : Continuous f := by
-  refine LowerTopology.continuous_of_Ici fun b => ?_
+protected lemma _root_.sInfHom.continuous (f : sInfHom α β) : Continuous f := by
+  refine IsLower.continuous_of_Ici fun b => ?_
   convert isClosed_Ici (a := sInf <| f ⁻¹' Ici b)
   refine' Subset.antisymm (fun a => sInf_le) fun a ha => le_trans _ <|
     OrderHomClass.mono (f : α →o β) ha
@@ -478,34 +425,32 @@ protected theorem sInfHom.continuous (f : sInfHom α β) : Continuous f := by
 #align Inf_hom.continuous sInfHom.continuous
 
 -- see Note [lower instance priority]
-instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
+instance (priority := 90) IsLower.toContinuousInf : ContinuousInf α :=
   ⟨(infsInfHom : sInfHom (α × α) α).continuous⟩
-#align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
+#align lower_topology.to_has_continuous_inf Topology.IsLower.toContinuousInf
 
-end CompleteLattice_LowerTopology
+end CompleteLattice_IsLower
 
-section CompleteLattice_UpperTopology
+section CompleteLattice_IsUpper
 
-variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [UpperTopology α]
-  [TopologicalSpace β] [UpperTopology β]
+variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [IsUpper α]
+  [TopologicalSpace β] [IsUpper β]
 
-protected theorem sSupHom.continuous (f : sSupHom α β) : Continuous f :=
+protected lemma _root_.sSupHom.continuous (f : sSupHom α β) : Continuous f :=
   sInfHom.continuous (α := αᵒᵈ) (β := βᵒᵈ) (sSupHom.dual.toFun f)
 
 -- see Note [lower instance priority]
-instance (priority := 90) UpperTopology.continuousInf : ContinuousSup α :=
+instance (priority := 90) IsUpper.toContinuousInf : ContinuousSup α :=
   ⟨(supsSupHom : sSupHom (α × α) α).continuous⟩
 
-end CompleteLattice_UpperTopology
+end CompleteLattice_IsUpper
 
-lemma upper_dual_iff_lower [Preorder α] [TopologicalSpace α] :
-    UpperTopology αᵒᵈ ↔ LowerTopology α := by
+lemma isUpper_orderDual [Preorder α] [TopologicalSpace α] : IsUpper αᵒᵈ ↔ IsLower α := by
   constructor
-  · apply UpperTopology.instLowerTopologyDual
-  · apply LowerTopology.instUpperTopologyDual
+  · apply OrderDual.instIsLower
+  · apply OrderDual.instIsUpper
 
-lemma lower_dual_iff_upper [Preorder α] [TopologicalSpace α] :
-    LowerTopology αᵒᵈ ↔ UpperTopology α := by
-  constructor
-  · apply LowerTopology.instUpperTopologyDual
-  · apply UpperTopology.instLowerTopologyDual
+lemma isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ ↔ IsUpper α :=
+  isUpper_orderDual.symm
+
+end Topology

Modify lemma names in UpperLowerSetTopology to meet Mathlib4 conventions.

Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com>

@@ -236,7 +236,11 @@ instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
 instance [Preorder α] : UpperTopology (WithUpperTopology α) :=
   ⟨rfl⟩
 
-def toOrderDualHomeomorph [Preorder α] : WithLowerTopology α ≃ₜ WithUpperTopology αᵒᵈ where
+/--
+The lower topology is homeomorphic to the upper topology on the dual order
+-/
+def WithLowerTopology.toDualHomeomorph [Preorder α] : WithLowerTopology α ≃ₜ WithUpperTopology αᵒᵈ
+    where
   toFun := OrderDual.toDual
   invFun := OrderDual.ofDual
   left_inv := OrderDual.toDual_ofDual

We define Alexandrov-discrete spaces as topological spaces where the intersection of a family of open sets is open.

This PR only gives a minimal API because the goal is to ensure that lemma names like isOpen_sInter are free to use for AlexandrovDiscrete. The existing lemmas are getting prefixed by Set.Finite or suffixed by _of_finite.

@@ -286,7 +286,7 @@ instance : ClosedIciTopology α :=
 /-- The upper closure of a finite set is closed in the lower topology. -/
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by
   simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
-  exact isClosed_biUnion h fun _ _ => isClosed_Ici
+  exact h.isClosed_biUnion fun _ _ => isClosed_Ici
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 
 /-- Every set open in the lower topology is a lower set. -/
@@ -448,7 +448,7 @@ instance instLowerTopologyProd [Preorder α] [TopologicalSpace α] [LowerTopolog
     simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]
     -- without `let`, `refine` tries to use the product topology and fails
     let _ : TopologicalSpace (α × β) := generateFrom { s | ∃ a, (Ici a)ᶜ = s }
-    refine (isOpen_biInter hs fun a _ => ?_).inter (isOpen_biInter ht fun b _ => ?_)
+    refine (hs.isOpen_biInter fun a _ => ?_).inter (ht.isOpen_biInter fun b _ => ?_)
     · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
 

@@ -494,13 +494,13 @@ instance (priority := 90) UpperTopology.continuousInf : ContinuousSup α :=
 
 end CompleteLattice_UpperTopology
 
-lemma UpperDual_iff_Lower [Preorder α] [TopologicalSpace α] :
+lemma upper_dual_iff_lower [Preorder α] [TopologicalSpace α] :
     UpperTopology αᵒᵈ ↔ LowerTopology α := by
   constructor
   · apply UpperTopology.instLowerTopologyDual
   · apply LowerTopology.instUpperTopologyDual
 
-lemma LowerDual_iff_Upper [Preorder α] [TopologicalSpace α] :
+lemma lower_dual_iff_upper [Preorder α] [TopologicalSpace α] :
     LowerTopology αᵒᵈ ↔ UpperTopology α := by
   constructor
   · apply LowerTopology.instUpperTopologyDual

As discussed a while ago on Zulip, we introduce classes expressing that a certain topology has closed Icis/Iics, which is sufficient to get boundedness of compacts on the desired side. The main application is that these now apply to types satisfying UpperTopology/LowerTopology, which will allow us to apply these compactness results to semicontinuous functions.

The naming was discussed here

@@ -276,14 +276,17 @@ instance instUpperTopologyDual [Preorder α] [TopologicalSpace α] [LowerTopolog
   topology_eq_upperTopology := topology_eq_lowerTopology (α := α)
 
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
-theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
-  isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
-#align lower_topology.is_closed_Ici LowerTopology.isClosed_Ici
+instance : ClosedIciTopology α :=
+  ⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩
+
+-- Porting note: The old `LowerTopology.isClosed_Ici` was removed, since one can now use
+-- the general `isClosed_Ici` lemma thanks to the instance above.
+#align lower_topology.is_closed_Ici isClosed_Ici
 
 /-- The upper closure of a finite set is closed in the lower topology. -/
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by
   simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
-  exact isClosed_biUnion h fun a _ => isClosed_Ici a
+  exact isClosed_biUnion h fun _ _ => isClosed_Ici
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 
 /-- Every set open in the lower topology is a lower set. -/
@@ -307,7 +310,7 @@ The closure of a singleton `{a}` in the lower topology is the left-closed right-
 -/
 @[simp]
 theorem closure_singleton (a : α) : closure {a} = Ici a :=
-  Subset.antisymm ((closure_minimal fun _ h => h.ge) <| isClosed_Ici a) <|
+  Subset.antisymm ((closure_minimal fun _ h => h.ge) <| isClosed_Ici) <|
     (isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl
 #align lower_topology.closure_singleton LowerTopology.closure_singleton
 
@@ -383,8 +386,8 @@ instance instLowerTopologyDual [Preorder α] [TopologicalSpace α] [UpperTopolog
   topology_eq_lowerTopology := topology_eq_upperTopology (α := α)
 
 /-- Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/
-theorem isClosed_Iic (a : α) : IsClosed (Iic a) :=
-  isOpen_compl_iff.1 <| isOpen_iff_generate_Iic_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
+instance : ClosedIicTopology α :=
+  ⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Iic_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩
 
 /-- The lower closure of a finite set is closed in the upper topology. -/
 theorem isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) :=
@@ -437,7 +440,7 @@ instance instLowerTopologyProd [Preorder α] [TopologicalSpace α] [LowerTopolog
   topology_eq_lowerTopology := by
     refine' le_antisymm (le_generateFrom _) _
     · rintro _ ⟨x, rfl⟩
-      exact ((LowerTopology.isClosed_Ici _).prod <| LowerTopology.isClosed_Ici _).isOpen_compl
+      exact (isClosed_Ici.prod isClosed_Ici).isOpen_compl
     rw [(LowerTopology.isTopologicalBasis.prod LowerTopology.isTopologicalBasis).eq_generateFrom,
       le_generateFrom_iff_subset_isOpen, image2_subset_iff]
     rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
@@ -463,7 +466,7 @@ variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerT
 
 protected theorem sInfHom.continuous (f : sInfHom α β) : Continuous f := by
   refine LowerTopology.continuous_of_Ici fun b => ?_
-  convert LowerTopology.isClosed_Ici (sInf <| f ⁻¹' Ici b)
+  convert isClosed_Ici (a := sInf <| f ⁻¹' Ici b)
   refine' Subset.antisymm (fun a => sInf_le) fun a ha => le_trans _ <|
     OrderHomClass.mono (f : α →o β) ha
   refine' LE.le.trans _ (map_sInf f _).ge

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -51,6 +51,8 @@ of the lower topology to the spectrum of a complete lattice coincides with the h
 lower topology, upper topology, preorder
 -/
 
+set_option autoImplicit true
+
 open Set TopologicalSpace
 
 section WithLowerTopology

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -55,7 +55,7 @@ open Set TopologicalSpace
 
 section WithLowerTopology
 
-variable (α β : Type _)
+variable (α β : Type*)
 
 /-- Type synonym for a preorder equipped with the lower topology. -/
 def WithLowerTopology := α
@@ -106,7 +106,7 @@ theorem ofLower_inj {a b : WithLowerTopology α} : ofLower a = ofLower b ↔ a =
 #align with_lower_topology.of_lower_inj WithLowerTopology.ofLower_inj
 
 /-- A recursor for `WithLowerTopology`. Use as `induction x using WithLowerTopology.rec`. -/
-protected def rec {β : WithLowerTopology α → Sort _} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
+protected def rec {β : WithLowerTopology α → Sort*} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
   h (ofLower a)
 #align with_lower_topology.rec WithLowerTopology.rec
 
@@ -142,7 +142,7 @@ end WithLowerTopology
 
 section WithUpperTopology
 
-variable (α β : Type _)
+variable (α β : Type*)
 
 /-- Type synonym for a preorder equipped with the upper topology. -/
 def WithUpperTopology := α
@@ -182,7 +182,7 @@ theorem ofUpper_inj {a b : WithUpperTopology α} : ofUpper a = ofUpper b ↔ a =
   Iff.rfl
 
 /-- A recursor for `WithUpperTopology`. Use as `induction x using WithUpperTopology.rec`. -/
-protected def rec {β : WithUpperTopology α → Sort _} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
+protected def rec {β : WithUpperTopology α → Sort*} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
   h (ofUpper a)
 
 instance [Nonempty α] : Nonempty (WithUpperTopology α) :=
@@ -217,7 +217,7 @@ end WithUpperTopology
 The lower topology is the topology generated by the complements of the left-closed right-infinite
 intervals.
 -/
-class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
+class LowerTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
   topology_eq_lowerTopology : t = generateFrom { s | ∃ a, (Ici a)ᶜ = s }
 #align lower_topology LowerTopology
 
@@ -225,7 +225,7 @@ class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop
 The upper topology is the topology generated by the complements of the right-closed left-infinite
 intervals.
 -/
-class UpperTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
+class UpperTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
   topology_eq_upperTopology : t = generateFrom { s | ∃ a, (Iic a)ᶜ = s }
 
 instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
@@ -246,7 +246,7 @@ namespace LowerTopology
 
 /-- The complements of the upper closures of finite sets are a collection of lower sets
 which form a basis for the lower topology. -/
-def lowerBasis (α : Type _) [Preorder α] :=
+def lowerBasis (α : Type*) [Preorder α] :=
   { s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s }
 #align lower_topology.lower_basis LowerTopology.lowerBasis
 
@@ -356,7 +356,7 @@ namespace UpperTopology
 
 /-- The complements of the lower closures of finite sets are a collection of upper sets
 which form a basis for the upper topology. -/
-def upperBasis (α : Type _) [Preorder α] :=
+def upperBasis (α : Type*) [Preorder α] :=
   { s : Set α | ∃ t : Set α, t.Finite ∧ (lowerClosure t : Set α)ᶜ = s }
 
 section Preorder

Defines the upper topology, which is the dual of the lower topology on a preorder.

Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com> Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com>

@@ -10,10 +10,13 @@ import Mathlib.Order.Hom.CompleteLattice
 #align_import topology.order.lower_topology from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
 
 /-!
-# Lower topology
+# Lower and Upper topology
 
 This file introduces the lower topology on a preorder as the topology generated by the complements
-of the closed intervals to infinity.
+of the left-closed right-infinite intervals.
+
+For completeness we also introduce the dual upper topology, generated by the complements of the
+right-closed left-infinite intervals.
 
 ## Main statements
 
@@ -32,6 +35,8 @@ We define a mixin class `LowerTopology` for the class of types which are both a
 topology and where the topology is generated by the complements of the closed intervals to infinity.
 It is shown that `WithLowerTopology α` is an instance of `LowerTopology`.
 
+Similarly for the upper topology.
+
 ## Motivation
 
 The lower topology is used with the `Scott` topology to define the Lawson topology. The restriction
@@ -43,13 +48,14 @@ of the lower topology to the spectrum of a complete lattice coincides with the h
 
 ## Tags
 
-lower topology, preorder
+lower topology, upper topology, preorder
 -/
 
+open Set TopologicalSpace
 
-variable (α β : Type _)
+section WithLowerTopology
 
-open Set TopologicalSpace
+variable (α β : Type _)
 
 /-- Type synonym for a preorder equipped with the lower topology. -/
 def WithLowerTopology := α
@@ -132,16 +138,110 @@ theorem isOpen_def (T : Set (WithLowerTopology α)) :
 
 end WithLowerTopology
 
+end WithLowerTopology
+
+section WithUpperTopology
+
+variable (α β : Type _)
+
+/-- Type synonym for a preorder equipped with the upper topology. -/
+def WithUpperTopology := α
+
+variable {α β}
+
+namespace WithUpperTopology
+
+/-- `toUpper` is the identity function to the `WithUpperTopology` of a type.  -/
+@[match_pattern]
+def toUpper : α ≃ WithUpperTopology α := Equiv.refl _
+
+/-- `ofUpper` is the identity function from the `WithUpperTopology` of a type.  -/
+@[match_pattern]
+def ofUpper : WithUpperTopology α ≃ α := Equiv.refl _
+
+@[simp]
+theorem to_withUpperTopology_symm_eq {α} : (@toUpper α).symm = ofUpper :=
+  rfl
+
+@[simp]
+theorem of_withUpperTopology_symm_eq : (@ofUpper α).symm = toUpper :=
+  rfl
+
+@[simp]
+theorem toUpper_ofUpper (a : WithUpperTopology α) : toUpper (ofUpper a) = a :=
+  rfl
+
+@[simp]
+theorem ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a :=
+  rfl
+
+theorem toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b :=
+  Iff.rfl
+
+theorem ofUpper_inj {a b : WithUpperTopology α} : ofUpper a = ofUpper b ↔ a = b :=
+  Iff.rfl
+
+/-- A recursor for `WithUpperTopology`. Use as `induction x using WithUpperTopology.rec`. -/
+protected def rec {β : WithUpperTopology α → Sort _} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
+  h (ofUpper a)
+
+instance [Nonempty α] : Nonempty (WithUpperTopology α) :=
+  ‹Nonempty α›
+
+instance [Inhabited α] : Inhabited (WithUpperTopology α) :=
+  ‹Inhabited α›
+
+variable [Preorder α]
+
+instance : Preorder (WithUpperTopology α) :=
+  ‹Preorder α›
+
+instance : TopologicalSpace (WithUpperTopology α) :=
+  generateFrom { s | ∃ a, (Iic a)ᶜ = s }
+
+theorem isOpen_preimage_ofUpper (S : Set α) :
+    IsOpen (WithUpperTopology.ofUpper ⁻¹' S) ↔
+      (generateFrom { s : Set α | ∃ a : α, (Iic a)ᶜ = s }).IsOpen S :=
+  Iff.rfl
+
+theorem isOpen_def (T : Set (WithUpperTopology α)) :
+    IsOpen T ↔ (generateFrom { s : Set α | ∃ a : α, (Iic a)ᶜ = s }).IsOpen
+    (WithUpperTopology.toUpper ⁻¹' T) :=
+  Iff.rfl
+
+end WithUpperTopology
+
+end WithUpperTopology
+
 /--
-The lower topology is the topology generated by the complements of the closed intervals to infinity.
+The lower topology is the topology generated by the complements of the left-closed right-infinite
+intervals.
 -/
 class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
   topology_eq_lowerTopology : t = generateFrom { s | ∃ a, (Ici a)ᶜ = s }
 #align lower_topology LowerTopology
 
+/--
+The upper topology is the topology generated by the complements of the right-closed left-infinite
+intervals.
+-/
+class UpperTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
+  topology_eq_upperTopology : t = generateFrom { s | ∃ a, (Iic a)ᶜ = s }
+
 instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
   ⟨rfl⟩
 
+instance [Preorder α] : UpperTopology (WithUpperTopology α) :=
+  ⟨rfl⟩
+
+def toOrderDualHomeomorph [Preorder α] : WithLowerTopology α ≃ₜ WithUpperTopology αᵒᵈ where
+  toFun := OrderDual.toDual
+  invFun := OrderDual.ofDual
+  left_inv := OrderDual.toDual_ofDual
+  right_inv := OrderDual.ofDual_toDual
+  continuous_toFun := continuous_coinduced_rng
+  continuous_invFun := continuous_coinduced_rng
+
 namespace LowerTopology
 
 /-- The complements of the upper closures of finite sets are a collection of lower sets
@@ -169,6 +269,10 @@ theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (
   rw [topology_eq α]; rfl
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
 
+instance instUpperTopologyDual [Preorder α] [TopologicalSpace α] [LowerTopology α] :
+    UpperTopology (αᵒᵈ) where
+  topology_eq_upperTopology := topology_eq_lowerTopology (α := α)
+
 /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
 theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
   isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
@@ -247,8 +351,87 @@ end PartialOrder
 
 end LowerTopology
 
-instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [Preorder β]
-    [TopologicalSpace β] [LowerTopology β] [OrderBot β] : LowerTopology (α × β) where
+
+namespace UpperTopology
+
+/-- The complements of the lower closures of finite sets are a collection of upper sets
+which form a basis for the upper topology. -/
+def upperBasis (α : Type _) [Preorder α] :=
+  { s : Set α | ∃ t : Set α, t.Finite ∧ (lowerClosure t : Set α)ᶜ = s }
+
+section Preorder
+
+variable (α)
+variable [Preorder α] [TopologicalSpace α] [UpperTopology α] {s : Set α}
+
+lemma topology_eq : ‹_› = generateFrom { s | ∃ a : α, (Iic a)ᶜ = s } := topology_eq_upperTopology
+
+variable {α}
+
+/-- If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpperTopology α`.
+-/
+def withUpperTopologyHomeomorph : WithUpperTopology α ≃ₜ α :=
+  WithUpperTopology.ofUpper.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
+
+theorem isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s := by
+  rw [topology_eq α]; rfl
+
+instance instLowerTopologyDual [Preorder α] [TopologicalSpace α] [UpperTopology α] :
+    LowerTopology (αᵒᵈ) where
+  topology_eq_lowerTopology := topology_eq_upperTopology (α := α)
+
+/-- Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/
+theorem isClosed_Iic (a : α) : IsClosed (Iic a) :=
+  isOpen_compl_iff.1 <| isOpen_iff_generate_Iic_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩
+
+/-- The lower closure of a finite set is closed in the upper topology. -/
+theorem isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) :=
+  LowerTopology.isClosed_upperClosure (α := αᵒᵈ) h
+
+/-- Every set open in the upper topology is a upper set. -/
+theorem isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s :=
+  LowerTopology.isLowerSet_of_isOpen (α := αᵒᵈ) h
+
+theorem isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s :=
+  isUpperSet_compl.1 <| isUpperSet_of_isOpen h.isOpen_compl
+
+/--
+The closure of a singleton `{a}` in the upper topology is the left-infinite right-closed interval
+(-∞,a].
+-/
+@[simp]
+theorem closure_singleton (a : α) : closure {a} = Iic a :=
+  LowerTopology.closure_singleton (α := αᵒᵈ) _
+
+protected theorem isTopologicalBasis : IsTopologicalBasis (upperBasis α) :=
+  LowerTopology.isTopologicalBasis (α := αᵒᵈ)
+
+/-- A function `f : β → α` with upper topology in the codomain is continuous provided that the
+preimage of every interval `Set.Iic a` is a closed set.
+
+TODO: upgrade to an `iff`. -/
+lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :
+    Continuous f :=
+  LowerTopology.continuous_of_Ici (α := αᵒᵈ) h
+
+end Preorder
+
+
+section PartialOrder
+
+variable [PartialOrder α] [TopologicalSpace α] [UpperTopology α]
+
+-- see Note [lower instance priority]
+/-- The upper topology on a partial order is T₀. -/
+instance (priority := 90) t0Space : T0Space α :=
+  LowerTopology.t0Space (α := αᵒᵈ)
+
+end PartialOrder
+
+end UpperTopology
+
+instance instLowerTopologyProd [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α]
+    [Preorder β] [TopologicalSpace β] [LowerTopology β] [OrderBot β] : LowerTopology (α × β) where
   topology_eq_lowerTopology := by
     refine' le_antisymm (le_generateFrom _) _
     · rintro _ ⟨x, rfl⟩
@@ -264,7 +447,14 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
     · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
 
-section CompleteLattice
+instance instUpperTopologyProd [Preorder α] [TopologicalSpace α] [UpperTopology α] [OrderTop α]
+    [Preorder β] [TopologicalSpace β] [UpperTopology β] [OrderTop β] : UpperTopology (α × β) where
+  topology_eq_upperTopology := by
+    suffices : LowerTopology (α × β)ᵒᵈ
+    · exact LowerTopology.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
+    exact instLowerTopologyProd (α := αᵒᵈ) (β := βᵒᵈ)
+
+section CompleteLattice_LowerTopology
 
 variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
   [TopologicalSpace β] [LowerTopology β]
@@ -283,4 +473,30 @@ instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
   ⟨(infsInfHom : sInfHom (α × α) α).continuous⟩
 #align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
 
-end CompleteLattice
+end CompleteLattice_LowerTopology
+
+section CompleteLattice_UpperTopology
+
+variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [UpperTopology α]
+  [TopologicalSpace β] [UpperTopology β]
+
+protected theorem sSupHom.continuous (f : sSupHom α β) : Continuous f :=
+  sInfHom.continuous (α := αᵒᵈ) (β := βᵒᵈ) (sSupHom.dual.toFun f)
+
+-- see Note [lower instance priority]
+instance (priority := 90) UpperTopology.continuousInf : ContinuousSup α :=
+  ⟨(supsSupHom : sSupHom (α × α) α).continuous⟩
+
+end CompleteLattice_UpperTopology
+
+lemma UpperDual_iff_Lower [Preorder α] [TopologicalSpace α] :
+    UpperTopology αᵒᵈ ↔ LowerTopology α := by
+  constructor
+  · apply UpperTopology.instLowerTopologyDual
+  · apply LowerTopology.instUpperTopologyDual
+
+lemma LowerDual_iff_Upper [Preorder α] [TopologicalSpace α] :
+    LowerTopology αᵒᵈ ↔ UpperTopology α := by
+  constructor
+  · apply LowerTopology.instUpperTopologyDual
+  · apply UpperTopology.instLowerTopologyDual

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Christopher Hoskin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
-
-! This file was ported from Lean 3 source module topology.order.lower_topology
-! leanprover-community/mathlib commit 98e83c3d541c77cdb7da20d79611a780ff8e7d90
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Homeomorph
 import Mathlib.Topology.Order.Lattice
 import Mathlib.Order.Hom.CompleteLattice
 
+#align_import topology.order.lower_topology from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
+
 /-!
 # Lower topology
 

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -119,17 +119,17 @@ instance : Preorder (WithLowerTopology α) :=
   ‹Preorder α›
 
 instance : TopologicalSpace (WithLowerTopology α) :=
-  generateFrom { s | ∃ a, Ici aᶜ = s }
+  generateFrom { s | ∃ a, (Ici a)ᶜ = s }
 
 theorem isOpen_preimage_ofLower (S : Set α) :
     IsOpen (WithLowerTopology.ofLower ⁻¹' S) ↔
-      (generateFrom { s : Set α | ∃ a : α, Ici aᶜ = s }).IsOpen S :=
+      (generateFrom { s : Set α | ∃ a : α, (Ici a)ᶜ = s }).IsOpen S :=
   Iff.rfl
 #align with_lower_topology.is_open_preimage_of_lower WithLowerTopology.isOpen_preimage_ofLower
 
 theorem isOpen_def (T : Set (WithLowerTopology α)) :
-    IsOpen T ↔
-      (generateFrom { s : Set α | ∃ a : α, Ici aᶜ = s }).IsOpen (WithLowerTopology.toLower ⁻¹' T) :=
+    IsOpen T ↔ (generateFrom { s : Set α | ∃ a : α, (Ici a)ᶜ = s }).IsOpen
+    (WithLowerTopology.toLower ⁻¹' T) :=
   Iff.rfl
 #align with_lower_topology.is_open_def WithLowerTopology.isOpen_def
 
@@ -139,7 +139,7 @@ end WithLowerTopology
 The lower topology is the topology generated by the complements of the closed intervals to infinity.
 -/
 class LowerTopology (α : Type _) [t : TopologicalSpace α] [Preorder α] : Prop where
-  topology_eq_lowerTopology : t = generateFrom { s | ∃ a, Ici aᶜ = s }
+  topology_eq_lowerTopology : t = generateFrom { s | ∃ a, (Ici a)ᶜ = s }
 #align lower_topology LowerTopology
 
 instance [Preorder α] : LowerTopology (WithLowerTopology α) :=
@@ -168,7 +168,7 @@ def withLowerTopologyHomeomorph : WithLowerTopology α ≃ₜ α :=
   WithLowerTopology.ofLower.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; rfl⟩
 #align lower_topology.with_lower_topology_homeomorph LowerTopology.withLowerTopologyHomeomorph
 
-theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, Ici aᶜ = t } s := by
+theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s := by
   rw [topology_eq α]; rfl
 #align lower_topology.is_open_iff_generate_Ici_compl LowerTopology.isOpen_iff_generate_Ici_compl
 
@@ -214,7 +214,7 @@ protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
   ext s
   constructor
   · rintro ⟨F, hF, rfl⟩
-    refine' ⟨(fun a => Ici aᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, _⟩
+    refine' ⟨(fun a => (Ici a)ᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, _⟩
     simp only [sInter_image]
   · rintro ⟨F, ⟨hF, hs⟩, rfl⟩
     haveI := hF.to_subtype
@@ -262,7 +262,7 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
     dsimp
     simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]
     -- without `let`, `refine` tries to use the product topology and fails
-    let _ : TopologicalSpace (α × β) := generateFrom { s | ∃ a, Ici aᶜ = s }
+    let _ : TopologicalSpace (α × β) := generateFrom { s | ∃ a, (Ici a)ᶜ = s }
     refine (isOpen_biInter hs fun a _ => ?_).inter (isOpen_biInter ht fun b _ => ?_)
     · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -179,8 +179,8 @@ theorem isClosed_Ici (a : α) : IsClosed (Ici a) :=
 
 /-- The upper closure of a finite set is closed in the lower topology. -/
 theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by
-  simp only [← UpperSet.infᵢ_Ici, UpperSet.coe_infᵢ]
-  exact isClosed_bunionᵢ h fun a _ => isClosed_Ici a
+  simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
+  exact isClosed_biUnion h fun a _ => isClosed_Ici a
 #align lower_topology.is_closed_upper_closure LowerTopology.isClosed_upperClosure
 
 /-- Every set open in the lower topology is a lower set. -/
@@ -191,7 +191,7 @@ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
   case basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl
   case univ => exact isLowerSet_univ
   case inter u v _ _ hu2 hv2 => exact hu2.inter hv2
-  case unionₛ _ _ ih => exact isLowerSet_unionₛ ih
+  case sUnion _ _ ih => exact isLowerSet_sUnion ih
 #align lower_topology.is_lower_set_of_is_open LowerTopology.isLowerSet_of_isOpen
 
 theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
@@ -210,17 +210,17 @@ theorem closure_singleton (a : α) : closure {a} = Ici a :=
 
 protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
   convert isTopologicalBasis_of_subbasis (topology_eq α)
-  simp_rw [lowerBasis, coe_upperClosure, compl_unionᵢ]
+  simp_rw [lowerBasis, coe_upperClosure, compl_iUnion]
   ext s
   constructor
   · rintro ⟨F, hF, rfl⟩
     refine' ⟨(fun a => Ici aᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, _⟩
-    simp only [interₛ_image]
+    simp only [sInter_image]
   · rintro ⟨F, ⟨hF, hs⟩, rfl⟩
     haveI := hF.to_subtype
     rw [subset_def, Subtype.forall'] at hs
     choose f hf using hs
-    exact ⟨_, finite_range f, by simp_rw [binterᵢ_range, hf, interₛ_eq_interᵢ]⟩
+    exact ⟨_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]⟩
 #align lower_topology.is_topological_basis LowerTopology.isTopologicalBasis
 
 /-- A function `f : β → α` with lower topology in the codomain is continuous provided that the
@@ -260,10 +260,10 @@ instance [Preorder α] [TopologicalSpace α] [LowerTopology α] [OrderBot α] [P
       le_generateFrom_iff_subset_isOpen, image2_subset_iff]
     rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
     dsimp
-    simp_rw [coe_upperClosure, compl_unionᵢ, prod_eq, preimage_interᵢ, preimage_compl]
+    simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]
     -- without `let`, `refine` tries to use the product topology and fails
     let _ : TopologicalSpace (α × β) := generateFrom { s | ∃ a, Ici aᶜ = s }
-    refine (isOpen_binterᵢ hs fun a _ => ?_).inter (isOpen_binterᵢ ht fun b _ => ?_)
+    refine (isOpen_biInter hs fun a _ => ?_).inter (isOpen_biInter ht fun b _ => ?_)
     · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
     · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩
 
@@ -272,19 +272,18 @@ section CompleteLattice
 variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [LowerTopology α]
   [TopologicalSpace β] [LowerTopology β]
 
-protected theorem InfₛHom.continuous (f : InfₛHom α β) : Continuous f := by
+protected theorem sInfHom.continuous (f : sInfHom α β) : Continuous f := by
   refine LowerTopology.continuous_of_Ici fun b => ?_
-  convert LowerTopology.isClosed_Ici (infₛ <| f ⁻¹' Ici b)
-  refine' Subset.antisymm (fun a => infₛ_le) fun a ha => le_trans _ <|
+  convert LowerTopology.isClosed_Ici (sInf <| f ⁻¹' Ici b)
+  refine' Subset.antisymm (fun a => sInf_le) fun a ha => le_trans _ <|
     OrderHomClass.mono (f : α →o β) ha
-  refine' LE.le.trans _ (map_infₛ f _).ge
+  refine' LE.le.trans _ (map_sInf f _).ge
   simp
-#align Inf_hom.continuous InfₛHom.continuous
+#align Inf_hom.continuous sInfHom.continuous
 
 -- see Note [lower instance priority]
 instance (priority := 90) LowerTopology.continuousInf : ContinuousInf α :=
-  ⟨(infInfₛHom : InfₛHom (α × α) α).continuous⟩
+  ⟨(infsInfHom : sInfHom (α × α) α).continuous⟩
 #align lower_topology.to_has_continuous_inf LowerTopology.continuousInf
 
 end CompleteLattice
-

Co-authored-by: Johan Commelin <johan@commelin.net>