topology.local_at_target ⟷ Mathlib.Topology.LocalAtTarget

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathbin.Topology.Sets.Opens
+import Topology.Sets.Opens
 
 #align_import topology.local_at_target from "leanprover-community/mathlib"@"25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e"
 

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

@@ -41,7 +41,7 @@ theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
 #align set.restrict_preimage_inducing Set.restrictPreimage_inducing
 -/
 
-alias Set.restrictPreimage_inducing ← Inducing.restrictPreimage
+alias Inducing.restrictPreimage := Set.restrictPreimage_inducing
 #align inducing.restrict_preimage Inducing.restrictPreimage
 
 #print Set.restrictPreimage_embedding /-
@@ -51,7 +51,7 @@ theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
 #align set.restrict_preimage_embedding Set.restrictPreimage_embedding
 -/
 
-alias Set.restrictPreimage_embedding ← Embedding.restrictPreimage
+alias Embedding.restrictPreimage := Set.restrictPreimage_embedding
 #align embedding.restrict_preimage Embedding.restrictPreimage
 
 #print Set.restrictPreimage_openEmbedding /-
@@ -62,7 +62,7 @@ theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
 #align set.restrict_preimage_open_embedding Set.restrictPreimage_openEmbedding
 -/
 
-alias Set.restrictPreimage_openEmbedding ← OpenEmbedding.restrictPreimage
+alias OpenEmbedding.restrictPreimage := Set.restrictPreimage_openEmbedding
 #align open_embedding.restrict_preimage OpenEmbedding.restrictPreimage
 
 #print Set.restrictPreimage_closedEmbedding /-
@@ -73,7 +73,7 @@ theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f
 #align set.restrict_preimage_closed_embedding Set.restrictPreimage_closedEmbedding
 -/
 
-alias Set.restrictPreimage_closedEmbedding ← ClosedEmbedding.restrictPreimage
+alias ClosedEmbedding.restrictPreimage := Set.restrictPreimage_closedEmbedding
 #align closed_embedding.restrict_preimage ClosedEmbedding.restrictPreimage
 
 #print Set.restrictPreimage_isClosedMap /-

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module topology.local_at_target
-! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Sets.Opens
 
+#align_import topology.local_at_target from "leanprover-community/mathlib"@"25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e"
+
 /-!
 # Properties of maps that are local at the target.
 

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

@@ -94,8 +94,7 @@ theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
 #align set.restrict_preimage_is_closed_map Set.restrictPreimage_isClosedMap
 -/
 
-include hU
-
+#print isOpen_iff_inter_of_iSup_eq_top /-
 theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) :=
   by
   constructor
@@ -105,7 +104,9 @@ theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpe
     rw [← s.inter_univ, ← this, Set.inter_iUnion]
     exact isOpen_iUnion H
 #align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_top
+-/
 
+#print isOpen_iff_coe_preimage_of_iSup_eq_top /-
 theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsOpen s ↔ ∀ i, IsOpen (coe ⁻¹' s : Set (U i)) :=
   by
@@ -114,12 +115,16 @@ theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
   apply isOpen_iff_inter_of_iSup_eq_top
   assumption
 #align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_top
+-/
 
+#print isClosed_iff_coe_preimage_of_iSup_eq_top /-
 theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed (coe ⁻¹' s : Set (U i)) := by
   simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)
 #align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_top
+-/
 
+#print isClosedMap_iff_isClosedMap_of_iSup_eq_top /-
 theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) :=
   by
@@ -133,7 +138,9 @@ theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     simpa [Set.restrictPreimage, ← Subtype.coe_inj]
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, c.symm ▸ hx, b, c⟩, fun ⟨a, _, b, c⟩ => ⟨a, b, c⟩⟩
 #align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_top
+-/
 
+#print inducing_iff_inducing_of_iSup_eq_top /-
 theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
     Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) :=
   by
@@ -148,7 +155,9 @@ theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
       inf_eq_left, Filter.le_principal_iff]
     exact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)
 #align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_top
+-/
 
+#print embedding_iff_embedding_of_iSup_eq_top /-
 theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
     Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) :=
   by
@@ -158,7 +167,9 @@ theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
   · apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
   · apply Set.injective_iff_injective_of_iUnion_eq_univ; convert congr_arg coe hU; simp
 #align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_top
+-/
 
+#print openEmbedding_iff_openEmbedding_of_iSup_eq_top /-
 theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
     OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) :=
   by
@@ -168,7 +179,9 @@ theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
   · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]; apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
 #align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_top
+-/
 
+#print closedEmbedding_iff_closedEmbedding_of_iSup_eq_top /-
 theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
     ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) :=
   by
@@ -178,4 +191,5 @@ theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
   · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]; apply isClosed_iff_coe_preimage_of_iSup_eq_top hU
 #align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_top
+-/
 

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

@@ -38,7 +38,7 @@ theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
     Inducing (s.restrictPreimage f) :=
   by
   simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict,
-    restrict_eq, ← @Filter.comap_comap _ _ _ _ coe f] at h⊢
+    restrict_eq, ← @Filter.comap_comap _ _ _ _ coe f] at h ⊢
   intro a
   rw [← h, ← inducing_coe.nhds_eq_comap]
 #align set.restrict_preimage_inducing Set.restrictPreimage_inducing

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

@@ -27,7 +27,7 @@ We show that the following properties of continuous maps are local at the target
 
 open TopologicalSpace Set Filter
 
-open Topology Filter
+open scoped Topology Filter
 
 variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
 

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

@@ -96,12 +96,6 @@ theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
 
 include hU
 
-/- warning: is_open_iff_inter_of_supr_eq_top -> isOpen_iff_inter_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.hasInter.{u1} β) s ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)))) (U i)))))
-but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.instInterSet.{u1} β) s (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))))
-Case conversion may be inaccurate. Consider using '#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_topₓ'. -/
 theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) :=
   by
   constructor
@@ -112,12 +106,6 @@ theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpe
     exact isOpen_iUnion H
 #align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_top
 
-/- warning: is_open_iff_coe_preimage_of_supr_eq_top -> isOpen_iff_coe_preimage_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
-but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
-Case conversion may be inaccurate. Consider using '#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_topₓ'. -/
 theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsOpen s ↔ ∀ i, IsOpen (coe ⁻¹' s : Set (U i)) :=
   by
@@ -127,23 +115,11 @@ theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
   assumption
 #align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_top
 
-/- warning: is_closed_iff_coe_preimage_of_supr_eq_top -> isClosed_iff_coe_preimage_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
-but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
-Case conversion may be inaccurate. Consider using '#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_topₓ'. -/
 theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed (coe ⁻¹' s : Set (U i)) := by
   simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)
 #align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_top
 
-/- warning: is_closed_map_iff_is_closed_map_of_supr_eq_top -> isClosedMap_iff_isClosedMap_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Iff (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Iff (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_topₓ'. -/
 theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) :=
   by
@@ -158,12 +134,6 @@ theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, c.symm ▸ hx, b, c⟩, fun ⟨a, _, b, c⟩ => ⟨a, b, c⟩⟩
 #align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_top
 
-/- warning: inducing_iff_inducing_of_supr_eq_top -> inducing_iff_inducing_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_topₓ'. -/
 theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
     Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) :=
   by
@@ -179,12 +149,6 @@ theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
     exact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)
 #align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_top
 
-/- warning: embedding_iff_embedding_of_supr_eq_top -> embedding_iff_embedding_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_topₓ'. -/
 theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
     Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) :=
   by
@@ -195,12 +159,6 @@ theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
   · apply Set.injective_iff_injective_of_iUnion_eq_univ; convert congr_arg coe hU; simp
 #align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_top
 
-/- warning: open_embedding_iff_open_embedding_of_supr_eq_top -> openEmbedding_iff_openEmbedding_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_topₓ'. -/
 theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
     OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) :=
   by
@@ -211,12 +169,6 @@ theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
   · simp_rw [Set.range_restrictPreimage]; apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
 #align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_top
 
-/- warning: closed_embedding_iff_closed_embedding_of_supr_eq_top -> closedEmbedding_iff_closedEmbedding_of_iSup_eq_top is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_topₓ'. -/
 theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
     ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) :=
   by

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

@@ -107,10 +107,7 @@ theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpe
   constructor
   · exact fun H i => H.inter (U i).2
   · intro H
-    have : (⋃ i, (U i : Set β)) = Set.univ :=
-      by
-      convert congr_arg coe hU
-      simp
+    have : (⋃ i, (U i : Set β)) = Set.univ := by convert congr_arg coe hU; simp
     rw [← s.inter_univ, ← this, Set.inter_iUnion]
     exact isOpen_iUnion H
 #align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_top
@@ -173,14 +170,9 @@ theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
   simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict,
     restrict_eq, ← @Filter.comap_comap _ _ _ _ coe f]
   constructor
-  · intro H i x
-    rw [← H, ← inducing_coe.nhds_eq_comap]
+  · intro H i x; rw [← H, ← inducing_coe.nhds_eq_comap]
   · intro H x
-    obtain ⟨i, hi⟩ :=
-      opens.mem_supr.mp
-        (show f x ∈ iSup U by
-          rw [hU]
-          triv)
+    obtain ⟨i, hi⟩ := opens.mem_supr.mp (show f x ∈ iSup U by rw [hU]; triv)
     erw [← OpenEmbedding.map_nhds_eq (h.1 _ (U i).2).openEmbedding_subtype_val ⟨x, hi⟩]
     rw [(H i) ⟨x, hi⟩, Filter.subtype_coe_map_comap, Function.comp_apply, Subtype.coe_mk,
       inf_eq_left, Filter.le_principal_iff]
@@ -200,9 +192,7 @@ theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
   rw [forall_and]
   apply and_congr
   · apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
-  · apply Set.injective_iff_injective_of_iUnion_eq_univ
-    convert congr_arg coe hU
-    simp
+  · apply Set.injective_iff_injective_of_iUnion_eq_univ; convert congr_arg coe hU; simp
 #align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_top
 
 /- warning: open_embedding_iff_open_embedding_of_supr_eq_top -> openEmbedding_iff_openEmbedding_of_iSup_eq_top is a dubious translation:
@@ -218,8 +208,7 @@ theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
   rw [forall_and]
   apply and_congr
   · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
-  · simp_rw [Set.range_restrictPreimage]
-    apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
+  · simp_rw [Set.range_restrictPreimage]; apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
 #align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_top
 
 /- warning: closed_embedding_iff_closed_embedding_of_supr_eq_top -> closedEmbedding_iff_closedEmbedding_of_iSup_eq_top is a dubious translation:
@@ -235,7 +224,6 @@ theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
   rw [forall_and]
   apply and_congr
   · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
-  · simp_rw [Set.range_restrictPreimage]
-    apply isClosed_iff_coe_preimage_of_iSup_eq_top hU
+  · simp_rw [Set.range_restrictPreimage]; apply isClosed_iff_coe_preimage_of_iSup_eq_top hU
 #align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_top
 

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

@@ -31,7 +31,7 @@ open Topology Filter
 
 variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
 
-variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : supᵢ U = ⊤)
+variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : iSup U = ⊤)
 
 #print Set.restrictPreimage_inducing /-
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
@@ -96,13 +96,13 @@ theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
 
 include hU
 
-/- warning: is_open_iff_inter_of_supr_eq_top -> isOpen_iff_inter_of_supᵢ_eq_top is a dubious translation:
+/- warning: is_open_iff_inter_of_supr_eq_top -> isOpen_iff_inter_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.hasInter.{u1} β) s ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)))) (U i)))))
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.hasInter.{u1} β) s ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)))) (U i)))))
 but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.instInterSet.{u1} β) s (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))))
-Case conversion may be inaccurate. Consider using '#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_supᵢ_eq_topₓ'. -/
-theorem isOpen_iff_inter_of_supᵢ_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) :=
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.instInterSet.{u1} β) s (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))))
+Case conversion may be inaccurate. Consider using '#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_topₓ'. -/
+theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) :=
   by
   constructor
   · exact fun H i => H.inter (U i).2
@@ -111,63 +111,63 @@ theorem isOpen_iff_inter_of_supᵢ_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsO
       by
       convert congr_arg coe hU
       simp
-    rw [← s.inter_univ, ← this, Set.inter_unionᵢ]
-    exact isOpen_unionᵢ H
-#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_supᵢ_eq_top
+    rw [← s.inter_univ, ← this, Set.inter_iUnion]
+    exact isOpen_iUnion H
+#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_top
 
-/- warning: is_open_iff_coe_preimage_of_supr_eq_top -> isOpen_iff_coe_preimage_of_supᵢ_eq_top is a dubious translation:
+/- warning: is_open_iff_coe_preimage_of_supr_eq_top -> isOpen_iff_coe_preimage_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
 but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
-Case conversion may be inaccurate. Consider using '#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_supᵢ_eq_topₓ'. -/
-theorem isOpen_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
+Case conversion may be inaccurate. Consider using '#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_topₓ'. -/
+theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsOpen s ↔ ∀ i, IsOpen (coe ⁻¹' s : Set (U i)) :=
   by
   simp_rw [(U _).2.openEmbedding_subtype_val.open_iff_image_open, Set.image_preimage_eq_inter_range,
     Subtype.range_coe]
-  apply isOpen_iff_inter_of_supᵢ_eq_top
+  apply isOpen_iff_inter_of_iSup_eq_top
   assumption
-#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_supᵢ_eq_top
+#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_top
 
-/- warning: is_closed_iff_coe_preimage_of_supr_eq_top -> isClosed_iff_coe_preimage_of_supᵢ_eq_top is a dubious translation:
+/- warning: is_closed_iff_coe_preimage_of_supr_eq_top -> isClosed_iff_coe_preimage_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
 but is expected to have type
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
-Case conversion may be inaccurate. Consider using '#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_supᵢ_eq_topₓ'. -/
-theorem isClosed_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
+Case conversion may be inaccurate. Consider using '#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_topₓ'. -/
+theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed (coe ⁻¹' s : Set (U i)) := by
-  simpa using isOpen_iff_coe_preimage_of_supᵢ_eq_top hU (sᶜ)
-#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_supᵢ_eq_top
+  simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)
+#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_top
 
-/- warning: is_closed_map_iff_is_closed_map_of_supr_eq_top -> isClosedMap_iff_isClosedMap_of_supᵢ_eq_top is a dubious translation:
+/- warning: is_closed_map_iff_is_closed_map_of_supr_eq_top -> isClosedMap_iff_isClosedMap_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Iff (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Iff (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Iff (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_supᵢ_eq_topₓ'. -/
-theorem isClosedMap_iff_isClosedMap_of_supᵢ_eq_top :
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Iff (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_topₓ'. -/
+theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) :=
   by
   refine' ⟨fun h i => Set.restrictPreimage_isClosedMap _ h, _⟩
   rintro H s hs
-  rw [isClosed_iff_coe_preimage_of_supᵢ_eq_top hU]
+  rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
   intro i
   convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩
   ext ⟨x, hx⟩
   suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, f y ∈ U i ∧ y ∈ s ∧ f y = x by
     simpa [Set.restrictPreimage, ← Subtype.coe_inj]
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, c.symm ▸ hx, b, c⟩, fun ⟨a, _, b, c⟩ => ⟨a, b, c⟩⟩
-#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_supᵢ_eq_top
+#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_top
 
-/- warning: inducing_iff_inducing_of_supr_eq_top -> inducing_iff_inducing_of_supᵢ_eq_top is a dubious translation:
+/- warning: inducing_iff_inducing_of_supr_eq_top -> inducing_iff_inducing_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_supᵢ_eq_topₓ'. -/
-theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_topₓ'. -/
+theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
     Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) :=
   by
   simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict,
@@ -178,64 +178,64 @@ theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
   · intro H x
     obtain ⟨i, hi⟩ :=
       opens.mem_supr.mp
-        (show f x ∈ supᵢ U by
+        (show f x ∈ iSup U by
           rw [hU]
           triv)
     erw [← OpenEmbedding.map_nhds_eq (h.1 _ (U i).2).openEmbedding_subtype_val ⟨x, hi⟩]
     rw [(H i) ⟨x, hi⟩, Filter.subtype_coe_map_comap, Function.comp_apply, Subtype.coe_mk,
       inf_eq_left, Filter.le_principal_iff]
     exact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)
-#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_supᵢ_eq_top
+#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_top
 
-/- warning: embedding_iff_embedding_of_supr_eq_top -> embedding_iff_embedding_of_supᵢ_eq_top is a dubious translation:
+/- warning: embedding_iff_embedding_of_supr_eq_top -> embedding_iff_embedding_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_supᵢ_eq_topₓ'. -/
-theorem embedding_iff_embedding_of_supᵢ_eq_top (h : Continuous f) :
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_topₓ'. -/
+theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
     Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) :=
   by
   simp_rw [embedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply inducing_iff_inducing_of_supᵢ_eq_top <;> assumption
-  · apply Set.injective_iff_injective_of_unionᵢ_eq_univ
+  · apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
+  · apply Set.injective_iff_injective_of_iUnion_eq_univ
     convert congr_arg coe hU
     simp
-#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_supᵢ_eq_top
+#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_top
 
-/- warning: open_embedding_iff_open_embedding_of_supr_eq_top -> openEmbedding_iff_openEmbedding_of_supᵢ_eq_top is a dubious translation:
+/- warning: open_embedding_iff_open_embedding_of_supr_eq_top -> openEmbedding_iff_openEmbedding_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_supᵢ_eq_topₓ'. -/
-theorem openEmbedding_iff_openEmbedding_of_supᵢ_eq_top (h : Continuous f) :
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_topₓ'. -/
+theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
     OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) :=
   by
   simp_rw [openEmbedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply embedding_iff_embedding_of_supᵢ_eq_top <;> assumption
+  · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]
-    apply isOpen_iff_coe_preimage_of_supᵢ_eq_top hU
-#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_supᵢ_eq_top
+    apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
+#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_top
 
-/- warning: closed_embedding_iff_closed_embedding_of_supr_eq_top -> closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top is a dubious translation:
+/- warning: closed_embedding_iff_closed_embedding_of_supr_eq_top -> closedEmbedding_iff_closedEmbedding_of_iSup_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (iSup.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
-Case conversion may be inaccurate. Consider using '#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_topₓ'. -/
-theorem closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top (h : Continuous f) :
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (iSup.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_topₓ'. -/
+theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
     ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) :=
   by
   simp_rw [closedEmbedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply embedding_iff_embedding_of_supᵢ_eq_top <;> assumption
+  · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]
-    apply isClosed_iff_coe_preimage_of_supᵢ_eq_top hU
-#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top
+    apply isClosed_iff_coe_preimage_of_iSup_eq_top hU
+#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_top
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module topology.local_at_target
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Topology.Sets.Opens
 /-!
 # Properties of maps that are local at the target.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We show that the following properties of continuous maps are local at the target :
 - `inducing`
 - `embedding`

mathlib commit https://github.com/leanprover-community/mathlib/commit/e05ead7993520a432bec94ac504842d90707ad63

@@ -30,6 +30,7 @@ variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α 
 
 variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : supᵢ U = ⊤)
 
+#print Set.restrictPreimage_inducing /-
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
     Inducing (s.restrictPreimage f) :=
   by
@@ -38,36 +39,44 @@ theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
   intro a
   rw [← h, ← inducing_coe.nhds_eq_comap]
 #align set.restrict_preimage_inducing Set.restrictPreimage_inducing
+-/
 
 alias Set.restrictPreimage_inducing ← Inducing.restrictPreimage
 #align inducing.restrict_preimage Inducing.restrictPreimage
 
+#print Set.restrictPreimage_embedding /-
 theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
     Embedding (s.restrictPreimage f) :=
   ⟨h.1.restrictPreimage s, h.2.restrictPreimage s⟩
 #align set.restrict_preimage_embedding Set.restrictPreimage_embedding
+-/
 
 alias Set.restrictPreimage_embedding ← Embedding.restrictPreimage
 #align embedding.restrict_preimage Embedding.restrictPreimage
 
+#print Set.restrictPreimage_openEmbedding /-
 theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
     OpenEmbedding (s.restrictPreimage f) :=
   ⟨h.1.restrictPreimage s,
     (s.range_restrictPreimage f).symm ▸ continuous_subtype_val.isOpen_preimage _ h.2⟩
 #align set.restrict_preimage_open_embedding Set.restrictPreimage_openEmbedding
+-/
 
 alias Set.restrictPreimage_openEmbedding ← OpenEmbedding.restrictPreimage
 #align open_embedding.restrict_preimage OpenEmbedding.restrictPreimage
 
+#print Set.restrictPreimage_closedEmbedding /-
 theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f) :
     ClosedEmbedding (s.restrictPreimage f) :=
   ⟨h.1.restrictPreimage s,
     (s.range_restrictPreimage f).symm ▸ inducing_subtype_val.isClosed_preimage _ h.2⟩
 #align set.restrict_preimage_closed_embedding Set.restrictPreimage_closedEmbedding
+-/
 
 alias Set.restrictPreimage_closedEmbedding ← ClosedEmbedding.restrictPreimage
 #align closed_embedding.restrict_preimage ClosedEmbedding.restrictPreimage
 
+#print Set.restrictPreimage_isClosedMap /-
 theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
     IsClosedMap (s.restrictPreimage f) :=
   by
@@ -80,9 +89,16 @@ theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
     simpa [Set.restrictPreimage, ← Subtype.coe_inj]
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, c.symm ▸ hx, b, c⟩, fun ⟨a, _, b, c⟩ => ⟨a, b, c⟩⟩
 #align set.restrict_preimage_is_closed_map Set.restrictPreimage_isClosedMap
+-/
 
 include hU
 
+/- warning: is_open_iff_inter_of_supr_eq_top -> isOpen_iff_inter_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.hasInter.{u1} β) s ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (Set.{u1} β) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)))) (U i)))))
+but is expected to have type
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} β _inst_2 (Inter.inter.{u1} (Set.{u1} β) (Set.instInterSet.{u1} β) s (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))))
+Case conversion may be inaccurate. Consider using '#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_supᵢ_eq_topₓ'. -/
 theorem isOpen_iff_inter_of_supᵢ_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) :=
   by
   constructor
@@ -96,6 +112,12 @@ theorem isOpen_iff_inter_of_supᵢ_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsO
     exact isOpen_unionᵢ H
 #align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_supᵢ_eq_top
 
+/- warning: is_open_iff_coe_preimage_of_supr_eq_top -> isOpen_iff_coe_preimage_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
+but is expected to have type
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (forall (i : ι), IsOpen.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
+Case conversion may be inaccurate. Consider using '#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_supᵢ_eq_topₓ'. -/
 theorem isOpen_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
     IsOpen s ↔ ∀ i, IsOpen (coe ⁻¹' s : Set (U i)) :=
   by
@@ -105,11 +127,23 @@ theorem isOpen_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
   assumption
 #align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_supᵢ_eq_top
 
+/- warning: is_closed_iff_coe_preimage_of_supr_eq_top -> isClosed_iff_coe_preimage_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) (Subtype.topologicalSpace.{u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (TopologicalSpace.Opens.{u1} β _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) (U i)) β (coeSubtype.{succ u1} β (fun (x : β) => Membership.Mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.setLike.{u1} β _inst_2)) x (U i))))))) s)))
+but is expected to have type
+  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {ι : Type.{u2}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u2} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (forall (s : Set.{u1} β), Iff (IsClosed.{u1} β _inst_2 s) (forall (i : ι), IsClosed.{u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i)) _inst_2) (Set.preimage.{u1, u1} (Subtype.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (TopologicalSpace.Opens.{u1} β _inst_2) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2)) x (U i))) β (Subtype.val.{succ u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} β _inst_2) β (TopologicalSpace.Opens.instSetLikeOpens.{u1} β _inst_2) (U i)))) s)))
+Case conversion may be inaccurate. Consider using '#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_supᵢ_eq_topₓ'. -/
 theorem isClosed_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed (coe ⁻¹' s : Set (U i)) := by
   simpa using isOpen_iff_coe_preimage_of_supᵢ_eq_top hU (sᶜ)
 #align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_supᵢ_eq_top
 
+/- warning: is_closed_map_iff_is_closed_map_of_supr_eq_top -> isClosedMap_iff_isClosedMap_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Iff (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Iff (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), IsClosedMap.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_supᵢ_eq_topₓ'. -/
 theorem isClosedMap_iff_isClosedMap_of_supᵢ_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) :=
   by
@@ -124,6 +158,12 @@ theorem isClosedMap_iff_isClosedMap_of_supᵢ_eq_top :
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, c.symm ▸ hx, b, c⟩, fun ⟨a, _, b, c⟩ => ⟨a, b, c⟩⟩
 #align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_supᵢ_eq_top
 
+/- warning: inducing_iff_inducing_of_supr_eq_top -> inducing_iff_inducing_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Inducing.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Inducing.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_supᵢ_eq_topₓ'. -/
 theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
     Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) :=
   by
@@ -144,6 +184,12 @@ theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
     exact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)
 #align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_supᵢ_eq_top
 
+/- warning: embedding_iff_embedding_of_supr_eq_top -> embedding_iff_embedding_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Embedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), Embedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_supᵢ_eq_topₓ'. -/
 theorem embedding_iff_embedding_of_supᵢ_eq_top (h : Continuous f) :
     Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) :=
   by
@@ -156,6 +202,12 @@ theorem embedding_iff_embedding_of_supᵢ_eq_top (h : Continuous f) :
     simp
 #align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_supᵢ_eq_top
 
+/- warning: open_embedding_iff_open_embedding_of_supr_eq_top -> openEmbedding_iff_openEmbedding_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (OpenEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), OpenEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_supᵢ_eq_topₓ'. -/
 theorem openEmbedding_iff_openEmbedding_of_supᵢ_eq_top (h : Continuous f) :
     OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) :=
   by
@@ -167,6 +219,12 @@ theorem openEmbedding_iff_openEmbedding_of_supᵢ_eq_top (h : Continuous f) :
     apply isOpen_iff_coe_preimage_of_supᵢ_eq_top hU
 #align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_supᵢ_eq_top
 
+/- warning: closed_embedding_iff_closed_embedding_of_supr_eq_top -> closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u2} β _inst_2)}, (Eq.{succ u2} (TopologicalSpace.Opens.{u2} β _inst_2) (supᵢ.{u2, succ u3} (TopologicalSpace.Opens.{u2} β _inst_2) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2))) ι U) (Top.top.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} β _inst_2) (TopologicalSpace.Opens.completeLattice.{u2} β _inst_2)))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.preimage.{u1, u2} α β f (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)))) _inst_1) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u1, u2} α β (TopologicalSpace.Opens.carrier.{u2} β _inst_2 (U i)) f)))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {U : ι -> (TopologicalSpace.Opens.{u1} β _inst_2)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} β _inst_2) (supᵢ.{u1, succ u3} (TopologicalSpace.Opens.{u1} β _inst_2) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2))) ι U) (Top.top.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} β _inst_2) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} β _inst_2)))) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (forall (i : ι), ClosedEmbedding.{u2, u1} (Set.Elem.{u2} α (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) (Set.Elem.{u1} β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.preimage.{u2, u1} α β f (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)))) _inst_1) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i))) _inst_2) (Set.restrictPreimage.{u2, u1} α β (TopologicalSpace.Opens.carrier.{u1} β _inst_2 (U i)) f)))
+Case conversion may be inaccurate. Consider using '#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_topₓ'. -/
 theorem closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top (h : Continuous f) :
     ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) :=
   by

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

• add the theorem image_restrictPreimage; use it to simplify the proof of Set.restrictPreimage_isClosedMap
• add the equivalent for open maps Set.restrictPreimage_isOpenMap.
• delete IsClosedMap.restrictPreimage, which duplicates Set.restrictPreimage_isClosedMap

Co-authored-by: BGuillemet <115193742+BGuillemet@users.noreply.github.com>

@@ -63,17 +63,29 @@ theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f
 alias ClosedEmbedding.restrictPreimage := Set.restrictPreimage_closedEmbedding
 #align closed_embedding.restrict_preimage ClosedEmbedding.restrictPreimage
 
-theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
+theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) :
     IsClosedMap (s.restrictPreimage f) := by
-  rintro t ⟨u, hu, e⟩
-  refine' ⟨⟨_, (H _ (IsOpen.isClosed_compl hu)).1, _⟩⟩
-  rw [← (congr_arg HasCompl.compl e).trans (compl_compl t)]
-  simp only [Set.preimage_compl, compl_inj_iff]
-  ext ⟨x, hx⟩
-  suffices (∃ y, y ∉ u ∧ f y = x) ↔ ∃ y, y ∉ u ∧ f y ∈ s ∧ f y = x by
-    simpa [Set.restrictPreimage, ← Subtype.coe_inj]
-  exact ⟨fun ⟨a, b, c⟩ => ⟨a, b, c.symm ▸ hx, c⟩, fun ⟨a, b, _, c⟩ => ⟨a, b, c⟩⟩
-#align set.restrict_preimage_is_closed_map Set.restrictPreimage_isClosedMap
+  intro t
+  suffices ∀ u, IsClosed u → Subtype.val ⁻¹' u = t →
+    ∃ v, IsClosed v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
+      simpa [isClosed_induced_iff]
+  exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
+
+@[deprecated] -- since 2024-04-02
+theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
+    IsClosedMap (s.restrictPreimage f) := H.restrictPreimage s
+
+theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
+    IsOpenMap (s.restrictPreimage f) := by
+  intro t
+  suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t →
+    ∃ v, IsOpen v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
+      simpa [isOpen_induced_iff]
+  exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
+
+@[deprecated] -- since 2024-04-02
+theorem Set.restrictPreimage_isOpenMap (s : Set β) (H : IsOpenMap f) :
+    IsOpenMap (s.restrictPreimage f) := H.restrictPreimage s
 
 theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by
   constructor
@@ -103,7 +115,7 @@ theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
 
 theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by
-  refine' ⟨fun h i => Set.restrictPreimage_isClosedMap _ h, _⟩
+  refine' ⟨fun h i => h.restrictPreimage _, _⟩
   rintro H s hs
   rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
   intro i

Std defines triv, a slight variation on trivial. It appears that Mathlib doesn't care about the distinction (any more?) and so we can consolidate on a single tactic.

https://github.com/leanprover/std4/pull/712 separately replaces triv in Std with an error explaining to use trivial.

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

@@ -126,7 +126,7 @@ theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
       Opens.mem_iSup.mp
         (show f x ∈ iSup U by
           rw [hU]
-          triv)
+          trivial)
     erw [← OpenEmbedding.map_nhds_eq (h.1 _ (U i).2).openEmbedding_subtype_val ⟨x, hi⟩]
     rw [(H i) ⟨x, hi⟩, Filter.subtype_coe_map_comap, Function.comp_apply, Subtype.coe_mk,
       inf_eq_left, Filter.le_principal_iff]

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -48,7 +48,7 @@ alias Embedding.restrictPreimage := Set.restrictPreimage_embedding
 theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
     OpenEmbedding (s.restrictPreimage f) :=
   ⟨h.1.restrictPreimage s,
-    (s.range_restrictPreimage f).symm ▸ continuous_subtype_val.isOpen_preimage _ h.2⟩
+    (s.range_restrictPreimage f).symm ▸ continuous_subtype_val.isOpen_preimage _ h.isOpen_range⟩
 #align set.restrict_preimage_open_embedding Set.restrictPreimage_openEmbedding
 
 alias OpenEmbedding.restrictPreimage := Set.restrictPreimage_openEmbedding
@@ -57,7 +57,7 @@ alias OpenEmbedding.restrictPreimage := Set.restrictPreimage_openEmbedding
 theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f) :
     ClosedEmbedding (s.restrictPreimage f) :=
   ⟨h.1.restrictPreimage s,
-    (s.range_restrictPreimage f).symm ▸ inducing_subtype_val.isClosed_preimage _ h.2⟩
+    (s.range_restrictPreimage f).symm ▸ inducing_subtype_val.isClosed_preimage _ h.isClosed_range⟩
 #align set.restrict_preimage_closed_embedding Set.restrictPreimage_closedEmbedding
 
 alias ClosedEmbedding.restrictPreimage := Set.restrictPreimage_closedEmbedding

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -24,7 +24,6 @@ open TopologicalSpace Set Filter
 open Topology Filter
 
 variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
-
 variable {s : Set β} {ι : Type*} {U : ι → Opens β} (hU : iSup U = ⊤)
 
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :

@@ -35,7 +35,7 @@ theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
   rw [← h, ← inducing_subtype_val.nhds_eq_comap]
 #align set.restrict_preimage_inducing Set.restrictPreimage_inducing
 
-alias Set.restrictPreimage_inducing ← Inducing.restrictPreimage
+alias Inducing.restrictPreimage := Set.restrictPreimage_inducing
 #align inducing.restrict_preimage Inducing.restrictPreimage
 
 theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
@@ -43,7 +43,7 @@ theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
   ⟨h.1.restrictPreimage s, h.2.restrictPreimage s⟩
 #align set.restrict_preimage_embedding Set.restrictPreimage_embedding
 
-alias Set.restrictPreimage_embedding ← Embedding.restrictPreimage
+alias Embedding.restrictPreimage := Set.restrictPreimage_embedding
 #align embedding.restrict_preimage Embedding.restrictPreimage
 
 theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
@@ -52,7 +52,7 @@ theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
     (s.range_restrictPreimage f).symm ▸ continuous_subtype_val.isOpen_preimage _ h.2⟩
 #align set.restrict_preimage_open_embedding Set.restrictPreimage_openEmbedding
 
-alias Set.restrictPreimage_openEmbedding ← OpenEmbedding.restrictPreimage
+alias OpenEmbedding.restrictPreimage := Set.restrictPreimage_openEmbedding
 #align open_embedding.restrict_preimage OpenEmbedding.restrictPreimage
 
 theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f) :
@@ -61,7 +61,7 @@ theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f
     (s.range_restrictPreimage f).symm ▸ inducing_subtype_val.isClosed_preimage _ h.2⟩
 #align set.restrict_preimage_closed_embedding Set.restrictPreimage_closedEmbedding
 
-alias Set.restrictPreimage_closedEmbedding ← ClosedEmbedding.restrictPreimage
+alias ClosedEmbedding.restrictPreimage := Set.restrictPreimage_closedEmbedding
 #align closed_embedding.restrict_preimage ClosedEmbedding.restrictPreimage
 
 theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :

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

This has nice performance benefits.

@@ -23,9 +23,9 @@ open TopologicalSpace Set Filter
 
 open Topology Filter
 
-variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
+variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
 
-variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : iSup U = ⊤)
+variable {s : Set β} {ι : Type*} {U : ι → Opens β} (hU : iSup U = ⊤)
 
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
     Inducing (s.restrictPreimage f) := by

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module topology.local_at_target
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Sets.Opens
 
+#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Properties of maps that are local at the target.
 

@@ -83,7 +83,7 @@ theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpe
   constructor
   · exact fun H i => H.inter (U i).2
   · intro H
-    have : (⋃ i, (U i : Set β)) = Set.univ := by
+    have : ⋃ i, (U i : Set β) = Set.univ := by
       convert congr_arg (SetLike.coe) hU
       simp
     rw [← s.inter_univ, ← this, Set.inter_iUnion]

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

@@ -102,7 +102,7 @@ theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
 
 theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by
-  simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)
+  simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU sᶜ
 #align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_top
 
 theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -33,7 +33,7 @@ variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : iSup U = ⊤)
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
     Inducing (s.restrictPreimage f) := by
   simp_rw [inducing_subtype_val.inducing_iff, inducing_iff_nhds, restrictPreimage,
-    MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h⊢
+    MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h ⊢
   intro a
   rw [← h, ← inducing_subtype_val.nhds_eq_comap]
 #align set.restrict_preimage_inducing Set.restrictPreimage_inducing

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -14,10 +14,10 @@ import Mathlib.Topology.Sets.Opens
 # Properties of maps that are local at the target.
 
 We show that the following properties of continuous maps are local at the target :
-- `inducing`
-- `embedding`
-- `open_embedding`
-- `closed_embedding`
+- `Inducing`
+- `Embedding`
+- `OpenEmbedding`
+- `ClosedEmbedding`
 
 -/
 

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>

@@ -28,7 +28,7 @@ open Topology Filter
 
 variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
 
-variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : supᵢ U = ⊤)
+variable {s : Set β} {ι : Type _} {U : ι → Opens β} (hU : iSup U = ⊤)
 
 theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
     Inducing (s.restrictPreimage f) := by
@@ -79,46 +79,46 @@ theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, b, c.symm ▸ hx, c⟩, fun ⟨a, b, _, c⟩ => ⟨a, b, c⟩⟩
 #align set.restrict_preimage_is_closed_map Set.restrictPreimage_isClosedMap
 
-theorem isOpen_iff_inter_of_supᵢ_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by
+theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by
   constructor
   · exact fun H i => H.inter (U i).2
   · intro H
     have : (⋃ i, (U i : Set β)) = Set.univ := by
       convert congr_arg (SetLike.coe) hU
       simp
-    rw [← s.inter_univ, ← this, Set.inter_unionᵢ]
-    exact isOpen_unionᵢ H
-#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_supᵢ_eq_top
+    rw [← s.inter_univ, ← this, Set.inter_iUnion]
+    exact isOpen_iUnion H
+#align is_open_iff_inter_of_supr_eq_top isOpen_iff_inter_of_iSup_eq_top
 
-theorem isOpen_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
+theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsOpen s ↔ ∀ i, IsOpen ((↑) ⁻¹' s : Set (U i)) := by
   -- Porting note: rewrote to avoid ´simp´ issues
-  rw [isOpen_iff_inter_of_supᵢ_eq_top hU s]
+  rw [isOpen_iff_inter_of_iSup_eq_top hU s]
   refine forall_congr' fun i => ?_
   rw [(U _).2.openEmbedding_subtype_val.open_iff_image_open]
   erw [Set.image_preimage_eq_inter_range]
   rw [Subtype.range_coe, Opens.carrier_eq_coe]
-#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_supᵢ_eq_top
+#align is_open_iff_coe_preimage_of_supr_eq_top isOpen_iff_coe_preimage_of_iSup_eq_top
 
-theorem isClosed_iff_coe_preimage_of_supᵢ_eq_top (s : Set β) :
+theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by
-  simpa using isOpen_iff_coe_preimage_of_supᵢ_eq_top hU (sᶜ)
-#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_supᵢ_eq_top
+  simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)
+#align is_closed_iff_coe_preimage_of_supr_eq_top isClosed_iff_coe_preimage_of_iSup_eq_top
 
-theorem isClosedMap_iff_isClosedMap_of_supᵢ_eq_top :
+theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
     IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by
   refine' ⟨fun h i => Set.restrictPreimage_isClosedMap _ h, _⟩
   rintro H s hs
-  rw [isClosed_iff_coe_preimage_of_supᵢ_eq_top hU]
+  rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
   intro i
   convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩
   ext ⟨x, hx⟩
   suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by
     simpa [Set.restrictPreimage, ← Subtype.coe_inj]
   exact ⟨fun ⟨a, b, c⟩ => ⟨a, b, c.symm ▸ hx, c⟩, fun ⟨a, b, _, c⟩ => ⟨a, b, c⟩⟩
-#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_supᵢ_eq_top
+#align is_closed_map_iff_is_closed_map_of_supr_eq_top isClosedMap_iff_isClosedMap_of_iSup_eq_top
 
-theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
+theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
     Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) := by
   simp_rw [inducing_subtype_val.inducing_iff, inducing_iff_nhds, restrictPreimage,
     MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f]
@@ -127,43 +127,43 @@ theorem inducing_iff_inducing_of_supᵢ_eq_top (h : Continuous f) :
     rw [Function.comp_apply, ← H, ← inducing_subtype_val.nhds_eq_comap]
   · intro H x
     obtain ⟨i, hi⟩ :=
-      Opens.mem_supᵢ.mp
-        (show f x ∈ supᵢ U by
+      Opens.mem_iSup.mp
+        (show f x ∈ iSup U by
           rw [hU]
           triv)
     erw [← OpenEmbedding.map_nhds_eq (h.1 _ (U i).2).openEmbedding_subtype_val ⟨x, hi⟩]
     rw [(H i) ⟨x, hi⟩, Filter.subtype_coe_map_comap, Function.comp_apply, Subtype.coe_mk,
       inf_eq_left, Filter.le_principal_iff]
     exact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)
-#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_supᵢ_eq_top
+#align inducing_iff_inducing_of_supr_eq_top inducing_iff_inducing_of_iSup_eq_top
 
-theorem embedding_iff_embedding_of_supᵢ_eq_top (h : Continuous f) :
+theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
     Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) := by
   simp_rw [embedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply inducing_iff_inducing_of_supᵢ_eq_top <;> assumption
-  · apply Set.injective_iff_injective_of_unionᵢ_eq_univ
+  · apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
+  · apply Set.injective_iff_injective_of_iUnion_eq_univ
     convert congr_arg SetLike.coe hU
     simp
-#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_supᵢ_eq_top
+#align embedding_iff_embedding_of_supr_eq_top embedding_iff_embedding_of_iSup_eq_top
 
-theorem openEmbedding_iff_openEmbedding_of_supᵢ_eq_top (h : Continuous f) :
+theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
     OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) := by
   simp_rw [openEmbedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply embedding_iff_embedding_of_supᵢ_eq_top <;> assumption
+  · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]
-    apply isOpen_iff_coe_preimage_of_supᵢ_eq_top hU
-#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_supᵢ_eq_top
+    apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
+#align open_embedding_iff_open_embedding_of_supr_eq_top openEmbedding_iff_openEmbedding_of_iSup_eq_top
 
-theorem closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top (h : Continuous f) :
+theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
     ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) := by
   simp_rw [closedEmbedding_iff]
   rw [forall_and]
   apply and_congr
-  · apply embedding_iff_embedding_of_supᵢ_eq_top <;> assumption
+  · apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
   · simp_rw [Set.range_restrictPreimage]
-    apply isClosed_iff_coe_preimage_of_supᵢ_eq_top hU
-#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_supᵢ_eq_top
+    apply isClosed_iff_coe_preimage_of_iSup_eq_top hU
+#align closed_embedding_iff_closed_embedding_of_supr_eq_top closedEmbedding_iff_closedEmbedding_of_iSup_eq_top

Co-authored-by: Moritz Doll <moritz.doll@googlemail.com>