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Result

Found 257 declarations mentioning Topology.IsInducing. Of these, only the first 200 are shown.

• Topology.IsInducing 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Prop
• Topology.IsEmbedding.toIsInducing 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (self : Topology.IsEmbedding f) : Topology.IsInducing f
• Topology.IsInducing.eq_induced 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (self : Topology.IsInducing f) : tX = TopologicalSpace.induced f tY
• Topology.IsInducing.mk 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (eq_induced : tX = TopologicalSpace.induced f tY) : Topology.IsInducing f
• Topology.isInducing_iff 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Topology.IsInducing f ↔ tX = TopologicalSpace.induced f tY
• Topology.IsEmbedding.mk 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (toIsInducing : Topology.IsInducing f) (injective : Function.Injective f) : Topology.IsEmbedding f
• Topology.isEmbedding_iff 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Topology.IsEmbedding f ↔ Topology.IsInducing f ∧ Function.Injective f
• Topology.IsInducing.id 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} [TopologicalSpace X] : Topology.IsInducing id
• Topology.IsInducing.induced 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] (f : X → Y) : Topology.IsInducing f
• Topology.IsInducing.of_subsingleton 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] [TopologicalSpace X] [Subsingleton X] (f : X → Y) : Topology.IsInducing f
• Topology.IsClosedEmbedding.isInducing 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsClosedEmbedding f) : Topology.IsInducing f
• Topology.IsEmbedding.isInducing 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) : Topology.IsInducing f
• Topology.IsInducing.continuous 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) : Continuous f
• Topology.IsInducing.indiscreteTopology 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] [TopologicalSpace X] [IndiscreteTopology Y] {f : X → Y} (hf : Topology.IsInducing f) : IndiscreteTopology X
• Topology.IsInducing.nontrivialTopology 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] [TopologicalSpace X] [NontrivialTopology X] {f : X → Y} (hf : Topology.IsInducing f) : NontrivialTopology Y
• Topology.IsOpenEmbedding.isInducing 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsOpenEmbedding f) : Topology.IsInducing f
• Topology.IsEmbedding.of_leftInverse_of_isInducing 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {g : Y → X} (h : Function.LeftInverse f g) (hf : Topology.IsInducing f) : Topology.IsEmbedding g
• Topology.IsInducing.isClosedMap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsInducing f) (h : IsClosed (Set.range f)) : IsClosedMap f
• Topology.IsInducing.isOpenMap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hi : Topology.IsInducing f) (ho : IsOpen (Set.range f)) : IsOpenMap f
• Topology.IsInducing.isClosed_preimage 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (h : Topology.IsInducing f) (s : Set Y) (hs : IsClosed s) : IsClosed (f ⁻¹' s)
• Topology.IsInducing.mapClusterPt_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {x : X} {l : Filter X} : MapClusterPt (f x) l f ↔ ClusterPt x l
• Topology.IsInducing.nhds_eq_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (x : X) : nhds x = Filter.comap f (nhds (f x))
• Topology.isInducing_iff_nhds 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] : Topology.IsInducing f ↔ ∀ (x : X), nhds x = Filter.comap f (nhds (f x))
• Topology.IsInducing.closure_eq_preimage_closure_image 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (s : Set X) : closure s = f ⁻¹' closure (f '' s)
• Topology.IsInducing.map_nhds_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (x : X) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)
• Topology.IsInducing.nhdsSet_eq_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (s : Set X) : nhdsSet s = Filter.comap f (nhdsSet (f '' s))
• Topology.IsInducing.comp 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : Topology.IsInducing g) (hf : Topology.IsInducing f) : Topology.IsInducing (g ∘ f)
• Topology.IsInducing.continuous_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : Topology.IsInducing g) : Continuous f ↔ Continuous (g ∘ f)
• Topology.IsInducing.of_comp_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : Topology.IsInducing g) : Topology.IsInducing (g ∘ f) ↔ Topology.IsInducing f
• Topology.IsInducing.dense_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} : Dense s ↔ ∀ (x : X), f x ∈ closure (f '' s)
• Topology.IsInducing.continuousAt_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : Topology.IsInducing g) {x : X} : ContinuousAt f x ↔ ContinuousAt (g ∘ f) x
• Topology.IsInducing.isClosed_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} : IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s
• Topology.IsInducing.isOpen_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} : IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s
• Topology.IsInducing.of_comp 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hf : Continuous f) (hg : Continuous g) (hgf : Topology.IsInducing (g ∘ f)) : Topology.IsInducing f
• Topology.IsInducing.setOfPred_isOpen 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) : {s | IsOpen s} = Set.preimage f '' {t | IsOpen t}
• Topology.IsInducing.setOf_isOpen 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) : {s | IsOpen s} = Set.preimage f '' {t | IsOpen t}
• Topology.IsInducing.tendsto_nhds_iff 📋 Mathlib.Topology.Maps.Basic
  {Y : Type u_2} {Z : Type u_3} {ι : Type u_4} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : Topology.IsInducing g) : Filter.Tendsto f l (nhds y) ↔ Filter.Tendsto (g ∘ f) l (nhds (g y))
• Topology.IsInducing.isClosed_iff' 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} : IsClosed s ↔ ∀ (x : X), f x ∈ closure (f '' s) → x ∈ s
• Topology.IsInducing.map_nhds_of_mem 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (x : X) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)
• Topology.IsInducing.image_eq_isClosed_inter_range 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} (hs : IsClosed s) : ∃ c, IsClosed c ∧ f '' s = c ∩ Set.range f
• Topology.IsInducing.image_eq_isOpen_inter_range 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {s : Set X} (hs : IsOpen s) : ∃ c, IsOpen c ∧ f '' s = c ∩ Set.range f
• Topology.IsInducing.basis_nhds 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {ι : Type u_4} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] {p : ι → Prop} {s : ι → Set Y} (hf : Topology.IsInducing f) {x : X} (h_basis : (nhds (f x)).HasBasis p s) : (nhds x).HasBasis p (Set.preimage f ∘ s)
• Topology.IsInducing.image_mem_nhdsWithin 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) {x : X} {s : Set X} (hs : s ∈ nhds x) : f '' s ∈ nhdsWithin (f x) (Set.range f)
• Topology.IsInducing.continuousAt_iff' 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hf : Topology.IsInducing f) {x : X} (h : Set.range f ∈ nhds (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
• Topology.IsInducing.isOpenQuotientMap_of_surjective 📋 Mathlib.Topology.Maps.OpenQuotient
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (ind : Topology.IsInducing f) (surj : Function.Surjective f) : IsOpenQuotientMap f
• Topology.IsInducing.isQuotientMap_of_surjective 📋 Mathlib.Topology.Maps.OpenQuotient
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (ind : Topology.IsInducing f) (surj : Function.Surjective f) : Topology.IsQuotientMap f
• isEmbedding_of_isOpenQuotientMap_of_isInducing 📋 Mathlib.Topology.Maps.OpenQuotient
  {A : Type u_4} {B : Type u_5} {C : Type u_6} {D : Type u_7} [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : A → B) (g : C → D) (p : A → C) (q : B → D) (h : g ∘ p = q ∘ f) (hf : Topology.IsInducing f) (hp : Topology.IsQuotientMap p) (hq : IsOpenQuotientMap q) (hg : Function.Injective g) (H : q ⁻¹' q '' Set.range f ⊆ Set.range f) : Topology.IsEmbedding g
• isQuotientMap_of_isOpenQuotientMap_of_isInducing 📋 Mathlib.Topology.Maps.OpenQuotient
  {A : Type u_4} {B : Type u_5} {C : Type u_6} {D : Type u_7} [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : A → B) (g : C → D) (p : A → C) (q : B → D) (h : g ∘ p = q ∘ f) (hf : Topology.IsInducing f) (hp : Function.Surjective p) (hq : IsOpenQuotientMap q) (hg : Topology.IsEmbedding g) (H : q ⁻¹' q '' Set.range f ⊆ Set.range f) : Topology.IsQuotientMap p
• coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing 📋 Mathlib.Topology.Maps.OpenQuotient
  {A : Type u_4} {B : Type u_5} {C : Type u_6} {D : Type u_7} [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace D] (f : A → B) (g : C → D) (p : A → C) (q : B → D) (h : g ∘ p = q ∘ f) (hf : Topology.IsInducing f) (hp : Function.Surjective p) (hq : IsOpenQuotientMap q) (hg : Function.Injective g) (H : q ⁻¹' q '' Set.range f ⊆ Set.range f) : TopologicalSpace.coinduced p inst✝ = TopologicalSpace.induced g inst✝¹
• Equiv.toHomeomorphOfIsInducing 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ Y) (hf : Topology.IsInducing ⇑f) : X ≃ₜ Y
• Homeomorph.isInducing 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Topology.IsInducing ⇑h
• Equiv.toEquiv_toHomeomorphOfIsInducing 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ Y) (hf : Topology.IsInducing ⇑f) : (f.toHomeomorphOfIsInducing hf).toEquiv = f
• Equiv.toHomeomorphOfIsInducing_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ Y) (hf : Topology.IsInducing ⇑f) : ⇑(f.toHomeomorphOfIsInducing hf) = ⇑f
• Equiv.toHomeomorphOfIsInducing_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ Y) (hf : Topology.IsInducing ⇑f) : ⇑(f.toHomeomorphOfIsInducing hf).symm = ⇑f.symm
• isInducing_prodMkRight 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X) : Topology.IsInducing (Prod.mk x)
• isInducing_prodMkLeft 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (y : Y) : Topology.IsInducing fun x => (x, y)
• Topology.IsInducing.sumSwap 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] : Topology.IsInducing Sum.swap
• Topology.IsInducing.sumElim_left 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} (h : Topology.IsInducing (Sum.elim f g)) : Topology.IsInducing f
• Topology.IsInducing.sumElim_right 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} (h : Topology.IsInducing (Sum.elim f g)) : Topology.IsInducing g
• Topology.isInducing_const_prod 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {x : X} {f : Y → Z} : (Topology.IsInducing fun x' => (x, f x')) ↔ Topology.IsInducing f
• Topology.isInducing_prod_const 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {y : Y} {f : X → Z} : (Topology.IsInducing fun x => (f x, y)) ↔ Topology.IsInducing f
• Topology.IsInducing.prodMap 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Y} {g : Z → W} (hf : Topology.IsInducing f) (hg : Topology.IsInducing g) : Topology.IsInducing (Prod.map f g)
• Topology.IsInducing.sumElim_of_separatedNhds 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} (hf : Topology.IsInducing f) (hg : Topology.IsInducing g) (hsep : SeparatedNhds (Set.range f) (Set.range g)) : Topology.IsInducing (Sum.elim f g)
• Topology.IsInducing.disjoint_of_sumElim_aux 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} (h : Topology.IsInducing (Sum.elim f g)) : Disjoint (closure (Set.range f)) (Set.range g)
• Topology.IsInducing.sumElim 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} (hf : Topology.IsInducing f) (hg : Topology.IsInducing g) (hFg : Disjoint (closure (Set.range f)) (Set.range g)) (hfG : Disjoint (Set.range f) (closure (Set.range g))) : Topology.IsInducing (Sum.elim f g)
• isInducing_sumElim 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {g : Y → Z} : Topology.IsInducing (Sum.elim f g) ↔ Topology.IsInducing f ∧ Topology.IsInducing g ∧ Disjoint (closure (Set.range f)) (Set.range g) ∧ Disjoint (Set.range f) (closure (Set.range g))
• Topology.IsInducing.subtypeVal 📋 Mathlib.Topology.Constructions
  {Y : Type v} [TopologicalSpace Y] {t : Set Y} : Topology.IsInducing Subtype.val
• Topology.IsInducing.codRestrict 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {e : X → Y} (he : Topology.IsInducing e) {s : Set Y} (hs : ∀ (x : X), e x ∈ s) : Topology.IsInducing (Set.codRestrict e s hs)
• Topology.IsInducing.of_codRestrict 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {t : Set Y} (ht : ∀ (x : X), f x ∈ t) (h : Topology.IsInducing (Set.codRestrict f t ht)) : Topology.IsInducing f
• Topology.IsInducing.piMap 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} {B : ι → Type u_4} [T : (i : ι) → TopologicalSpace (A i)] [(i : ι) → TopologicalSpace (B i)] {f : (i : ι) → A i → B i} (hf : ∀ (i : ι), Topology.IsInducing (f i)) : Topology.IsInducing (Pi.map f)
• inducing_iInf_to_pi 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] {X : Type u_6} (f : (i : ι) → X → A i) : Topology.IsInducing fun x i => f i x
• Topology.IsInducing.restrict 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) {s : Set X} {t : Set Y} (H : Set.MapsTo f s t) : Topology.IsInducing (Set.MapsTo.restrict f s t H)
• Topology.isInducing_sigmaMap 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {κ : Type u_3} {σ : ι → Type u_4} {τ : κ → Type u_5} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (h₁ : Function.Injective f₁) : Topology.IsInducing (Sigma.map f₁ f₂) ↔ ∀ (i : ι), Topology.IsInducing (f₂ i)
• inducing_sigma 📋 Mathlib.Topology.Constructions
  {X : Type u} {ι : Type u_2} {σ : ι → Type u_4} [(i : ι) → TopologicalSpace (σ i)] [TopologicalSpace X] {f : Sigma σ → X} : Topology.IsInducing f ↔ (∀ (i : ι), Topology.IsInducing (f ∘ Sigma.mk i)) ∧ ∀ (i : ι), ∃ U, IsOpen U ∧ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i
• Topology.IsInducing.map_nhdsSet_eq 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) (s : Set α) : Filter.map f (nhdsSet s) = nhdsSetWithin (f '' s) (Set.range f)
• Topology.IsInducing.map_nhdsWithin_eq 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) (s : Set α) (x : α) : Filter.map f (nhdsWithin x s) = nhdsWithin (f x) (f '' s)
• Topology.IsInducing.continuousOn_iff 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Topology.IsInducing g) {s : Set α} : ContinuousOn f s ↔ ContinuousOn (g ∘ f) s
• Topology.IsInducing.continuousWithinAt_iff 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Topology.IsInducing g) {s : Set α} {x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x
• Topology.IsInducing.continuousOn_image_iff 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : β → γ} {s : Set α} (hf : Topology.IsInducing f) : ContinuousOn g (f '' s) ↔ ContinuousOn (g ∘ f) s
• Topology.IsInducing.firstCountableTopology 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] {β : Type u_1} [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Topology.IsInducing f) : FirstCountableTopology α
• Topology.IsInducing.secondCountableTopology 📋 Mathlib.Topology.Bases
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : α → β} [TopologicalSpace β] [SecondCountableTopology β] (hf : Topology.IsInducing f) : SecondCountableTopology α
• IsOpenMap.separableSpace_of_isInducing 📋 Mathlib.Topology.Bases
  {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.SeparableSpace β] {f : α → β} (h : IsOpenMap f) (h' : Topology.IsInducing f) : TopologicalSpace.SeparableSpace α
• TopologicalSpace.IsTopologicalBasis.isInducing 📋 Mathlib.Topology.Bases
  {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace β] {f : α → β} {T : Set (Set β)} (hf : Topology.IsInducing f) (h : TopologicalSpace.IsTopologicalBasis T) : TopologicalSpace.IsTopologicalBasis (Set.preimage f '' T)
• Topology.IsInducing.isTopologicalBasis 📋 Mathlib.Topology.Bases
  {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) {T : Set (Set β)} (h : TopologicalSpace.IsTopologicalBasis T) : TopologicalSpace.IsTopologicalBasis (Set.preimage f '' T)
• Topology.IsInducing.isCompact_iff 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Topology.IsInducing f) : IsCompact s ↔ IsCompact (f '' s)
• Topology.IsInducing.isCompact_preimage 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) (hf' : IsClosed (Set.range f)) {K : Set Y} (hK : IsCompact K) : IsCompact (f ⁻¹' K)
• Topology.IsInducing.isCompact_preimage' 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) {K : Set Y} (hK : IsCompact K) (Kf : K ⊆ Set.range f) : IsCompact (f ⁻¹' K)
• Topology.IsInducing.isCompact_preimage_iff 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) {K : Set Y} (Kf : K ⊆ Set.range f) : IsCompact (f ⁻¹' K) ↔ IsCompact K
• Topology.IsInducing.locallyCompactSpace 📋 Mathlib.Topology.Compactness.LocallyCompact
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace Y] {f : X → Y} (hf : Topology.IsInducing f) (h : IsLocallyClosed (Set.range f)) : LocallyCompactSpace X
• Topology.IsInducing.isSigmaCompact_iff 📋 Mathlib.Topology.Compactness.SigmaCompact
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hf : Topology.IsInducing f) : IsSigmaCompact s ↔ IsSigmaCompact (f '' s)
• SeparationQuotient.isInducing_mk 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} [TopologicalSpace X] : Topology.IsInducing SeparationQuotient.mk
• Topology.IsInducing.generalizingMap 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) (h : StableUnderGeneralization (Set.range f)) : GeneralizingMap f
• Topology.IsInducing.specializingMap 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) (h : StableUnderSpecialization (Set.range f)) : SpecializingMap f
• Topology.IsInducing.inseparable_iff 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (hf : Topology.IsInducing f) : Inseparable (f x) (f y) ↔ Inseparable x y
• Topology.IsInducing.specializes_iff 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (hf : Topology.IsInducing f) : f x ⤳ f y ↔ x ⤳ y
• Topology.IsInducing.r0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R0Space X] [TopologicalSpace Y] {f : Y → X} (hf : Topology.IsInducing f) : R0Space Y
• Topology.IsInducing.r1Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R1Space X] [TopologicalSpace Y] {f : Y → X} (hf : Topology.IsInducing f) : R1Space Y
• Topology.IsInducing.injective 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Topology.IsInducing f) : Function.Injective f
• Topology.IsInducing.isEmbedding 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Topology.IsInducing f) : Topology.IsEmbedding f
• isEmbedding_iff_isInducing 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y} : Topology.IsEmbedding f ↔ Topology.IsInducing f
• IsEmbedding.isDiscrete_range 📋 Mathlib.Topology.DiscreteSubset
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [DiscreteTopology X] (hf : Topology.IsInducing f) : IsDiscrete (Set.range f)
• Topology.IsInducing.isDiscrete_range 📋 Mathlib.Topology.DiscreteSubset
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [DiscreteTopology X] (hf : Topology.IsInducing f) : IsDiscrete (Set.range f)
• IsDiscrete.image 📋 Mathlib.Topology.DiscreteSubset
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {s : Set X} [TopologicalSpace Y] {f : X → Y} (hs : IsDiscrete s) (hf : Topology.IsInducing f) : IsDiscrete (f '' s)
• Topology.IsInducing.isPreconnected_image 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : α → β} (hf : Topology.IsInducing f) : IsPreconnected (f '' s) ↔ IsPreconnected s
• Topology.IsInducing.isLindelof_iff 📋 Mathlib.Topology.Compactness.Lindelof
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Topology.IsInducing f) : IsLindelof s ↔ IsLindelof (f '' s)
• Topology.IsInducing.isLindelof_preimage 📋 Mathlib.Topology.Compactness.Lindelof
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) (hf' : IsClosed (Set.range f)) {K : Set Y} (hK : IsLindelof K) : IsLindelof (f ⁻¹' K)
• Topology.IsInducing.completelyNormalSpace 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompletelyNormalSpace Y] {e : X → Y} (he : Topology.IsInducing e) : CompletelyNormalSpace X
• Topology.IsInducing.regularSpace 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [RegularSpace X] [TopologicalSpace Y] {f : Y → X} (hf : Topology.IsInducing f) : RegularSpace Y
• IsDenseInducing.isInducing 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : IsDenseInducing i) : Topology.IsInducing i
• IsDenseInducing.toIsInducing 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (self : IsDenseInducing i) : Topology.IsInducing i
• IsDenseInducing.mk 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (toIsInducing : Topology.IsInducing i) (dense : DenseRange i) : IsDenseInducing i
• Topology.IsInducing.isMeagre_image 📋 Mathlib.Topology.GDelta.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) {s : Set X} (h : IsMeagre s) : IsMeagre (f '' s)
• Topology.IsInducing.isNowhereDense_image 📋 Mathlib.Topology.GDelta.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsInducing f) {s : Set X} (h : IsNowhereDense s) : IsNowhereDense (f '' s)
• IsHomeomorph.isInducing 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsHomeomorph f) : Topology.IsInducing f
• AddUnits.isInducing_embedProduct 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [AddMonoid M] : Topology.IsInducing ⇑(AddUnits.embedProduct M)
• Units.isInducing_embedProduct 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [Monoid M] : Topology.IsInducing ⇑(Units.embedProduct M)
• Topology.IsInducing.addUnits_map 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [AddMonoid M] [TopologicalSpace N] [AddMonoid N] {f : M →+ N} (hf : Topology.IsInducing ⇑f) : Topology.IsInducing ⇑(AddUnits.map f)
• Topology.IsInducing.units_map 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [Monoid M] [TopologicalSpace N] [Monoid N] {f : M →* N} (hf : Topology.IsInducing ⇑f) : Topology.IsInducing ⇑(Units.map f)
• Topology.IsInducing.continuousConstSMul 📋 Mathlib.Topology.Algebra.ConstMulAction
  {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] {N : Type u_4} {β : Type u_5} [SMul N β] [TopologicalSpace β] {g : β → α} (hg : Topology.IsInducing g) (f : N → M) (hf : ∀ {c : N} {x : β}, g (c • x) = f c • g x) : ContinuousConstSMul N β
• Topology.IsInducing.continuousConstVAdd 📋 Mathlib.Topology.Algebra.ConstMulAction
  {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] {N : Type u_4} {β : Type u_5} [VAdd N β] [TopologicalSpace β] {g : β → α} (hg : Topology.IsInducing g) (f : N → M) (hf : ∀ {c : N} {x : β}, g (c +ᵥ x) = f c +ᵥ g x) : ContinuousConstVAdd N β
• Topology.IsInducing.continuousSMul 📋 Mathlib.Topology.Algebra.MulAction
  {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {g : Y → X} {N : Type u_5} [SMul N Y] [TopologicalSpace N] {f : N → M} (hg : Topology.IsInducing g) (hf : Continuous f) (hsmul : ∀ {c : N} {x : Y}, g (c • x) = f c • g x) : ContinuousSMul N Y
• Topology.IsInducing.continuousVAdd 📋 Mathlib.Topology.Algebra.MulAction
  {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {g : Y → X} {N : Type u_5} [VAdd N Y] [TopologicalSpace N] {f : N → M} (hg : Topology.IsInducing g) (hf : Continuous f) (hvadd : ∀ {c : N} {x : Y}, g (c +ᵥ x) = f c +ᵥ g x) : ContinuousVAdd N Y
• Topology.IsInducing.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousAdd N] (f : F) (hf : Topology.IsInducing ⇑f) : ContinuousAdd M
• Topology.IsInducing.continuousMul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousMul N] (f : F) (hf : Topology.IsInducing ⇑f) : ContinuousMul M
• Topology.IsInducing.separatelyContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [SeparatelyContinuousAdd N] (f : F) (hf : Topology.IsInducing ⇑f) : SeparatelyContinuousAdd M
• Topology.IsInducing.separatelyContinuousMul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [SeparatelyContinuousMul N] (f : F) (hf : Topology.IsInducing ⇑f) : SeparatelyContinuousMul M
• IsUniformInducing.isInducing 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : IsUniformInducing f) : Topology.IsInducing f
• Topology.IsInducing.continuousInv 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} {H : Type u_5} [Inv G] [Inv H] [TopologicalSpace G] [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Topology.IsInducing f) (hf_inv : ∀ (x : G), f x⁻¹ = (f x)⁻¹) : ContinuousInv G
• Topology.IsInducing.continuousNeg 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} {H : Type u_5} [Neg G] [Neg H] [TopologicalSpace G] [TopologicalSpace H] [ContinuousNeg H] {f : G → H} (hf : Topology.IsInducing f) (hf_neg : ∀ (x : G), f (-x) = -f x) : ContinuousNeg G
• Topology.IsInducing.isTopologicalAddGroup 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {F : Type u_5} [AddGroup H] [TopologicalSpace H] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) (hf : Topology.IsInducing ⇑f) : IsTopologicalAddGroup H
• Topology.IsInducing.isTopologicalGroup 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] {F : Type u_5} [Group H] [TopologicalSpace H] [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Topology.IsInducing ⇑f) : IsTopologicalGroup H
• Topology.IsInducing.topologicalAddGroup 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {F : Type u_5} [AddGroup H] [TopologicalSpace H] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) (hf : Topology.IsInducing ⇑f) : IsTopologicalAddGroup H
• Topology.IsInducing.topologicalGroup 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] {F : Type u_5} [Group H] [TopologicalSpace H] [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Topology.IsInducing ⇑f) : IsTopologicalGroup H
• IsTopologicalAddGroup.isInducing_iff_nhds_zero 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] [IsTopologicalAddGroup H] {F : Type u_3} [FunLike F G H] [AddMonoidHomClass F G H] {f : F} : Topology.IsInducing ⇑f ↔ nhds 0 = Filter.comap (⇑f) (nhds 0)
• IsTopologicalGroup.isInducing_iff_nhds_one 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] {H : Type u_2} [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {F : Type u_3} [FunLike F G H] [MonoidHomClass F G H] {f : F} : Topology.IsInducing ⇑f ↔ nhds 1 = Filter.comap (⇑f) (nhds 1)
• AddMonoidHom.isUniformInducing_of_isInducing 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {Hom : Type u_3} [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [FunLike Hom α β] [AddMonoidHomClass Hom α β] {f : Hom} (h : Topology.IsInducing ⇑f) : IsUniformInducing ⇑f
• MonoidHom.isUniformInducing_of_isInducing 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {Hom : Type u_3} [UniformSpace β] [Group β] [IsUniformGroup β] [FunLike Hom α β] [MonoidHomClass Hom α β] {f : Hom} (h : Topology.IsInducing ⇑f) : IsUniformInducing ⇑f
• TopologicalSpace.Opens.IsBasis.of_isInducing 📋 Mathlib.Topology.Sets.Opens
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {B : Set (TopologicalSpace.Opens β)} (H : TopologicalSpace.Opens.IsBasis B) {f : α → β} (h : Topology.IsInducing f) : TopologicalSpace.Opens.IsBasis {x | ∃ U ∈ B, { carrier := f ⁻¹' ↑U, is_open' := ⋯ } = x}
• IsLocallyClosed.image 📋 Mathlib.Topology.LocallyClosed
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (hs : IsLocallyClosed s) {f : X → Y} (hf : Topology.IsInducing f) (hf' : IsLocallyClosed (Set.range f)) : IsLocallyClosed (f '' s)
• Topology.IsInducing.isLocallyClosed_iff 📋 Mathlib.Topology.LocallyClosed
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Topology.IsInducing f) : IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ f ⁻¹' s' = s
• Set.restrictPreimage_isInducing 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Topology.IsInducing f) : Topology.IsInducing (s.restrictPreimage f)
• Topology.IsInducing.restrictPreimage 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Topology.IsInducing f) : Topology.IsInducing (s.restrictPreimage f)
• TopologicalSpace.IsOpenCover.isInducing_iff_restrictPreimage 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : TopologicalSpace.IsOpenCover U) (h : Continuous f) : Topology.IsInducing f ↔ ∀ (i : ι), Topology.IsInducing ((U i).carrier.restrictPreimage f)
• TopologicalSpace.IrreducibleCloseds.map_injective_of_isInducing 📋 Mathlib.Topology.Sets.Closeds
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {f : β → α} (hf : Topology.IsInducing f) : Function.Injective (TopologicalSpace.IrreducibleCloseds.map f ⋯)
• TopologicalSpace.IrreducibleCloseds.map_strictMono_of_isInducing 📋 Mathlib.Topology.Sets.Closeds
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {f : β → α} (hf : Topology.IsInducing f) : StrictMono (TopologicalSpace.IrreducibleCloseds.map f ⋯)
• Topology.IsInducing.noetherianSpace 📋 Mathlib.Topology.NoetherianSpace
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.NoetherianSpace α] {i : β → α} (hi : Topology.IsInducing i) : TopologicalSpace.NoetherianSpace β
• TopologicalSpace.Compacts.range_map 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) : Set.range (TopologicalSpace.Compacts.map f ⋯) = {K | ↑K ⊆ Set.range f}
• TopologicalSpace.NonemptyCompacts.range_map 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) : Set.range (TopologicalSpace.NonemptyCompacts.map f ⋯) = {K | ↑K ⊆ Set.range f}
• PrespectralSpace.of_isInducing 📋 Mathlib.Topology.Spectral.Prespectral
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [PrespectralSpace Y] (f : X → Y) (hf : Topology.IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X
• Topology.IsInducing.topologicalKrullDim_le 📋 Mathlib.Topology.KrullDimension
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : Y → X} (hf : Topology.IsInducing f) : topologicalKrullDim Y ≤ topologicalKrullDim X
• PrimeSpectrum.localization_comap_isInducing 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] : Topology.IsInducing (PrimeSpectrum.comap (algebraMap R S))
• PrimeSpectrum.comap_isInducing_of_surjective 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) : Topology.IsInducing (PrimeSpectrum.comap f)
• TopCat.IsInducing.empty 📋 Mathlib.Topology.Category.TopCat.Limits.Basic
  (X : TopCat) : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom (TopCat.isInitialPEmpty.to X))
• TopCat.isInducing_prodMap 📋 Mathlib.Topology.Category.TopCat.Limits.Products
  {W X Y Z : TopCat} {f : W ⟶ X} {g : Y ⟶ Z} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (hg : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom g)) : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.prod.map f g))
• TopCat.isInducing_pullback_to_prod 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.pullback.fst f g) (CategoryTheory.Limits.pullback.snd f g)))
• Topology.IsInducing.functorObj 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f) → (U : TopologicalSpace.Opens ↑X) → TopologicalSpace.Opens ↑Y
• Topology.IsInducing.functor 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) : CategoryTheory.Functor (TopologicalSpace.Opens ↑X) (TopologicalSpace.Opens ↑Y)
• Topology.IsInducing.adjunction 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) : TopologicalSpace.Opens.map f ⊣ hf.functor
• Topology.IsInducing.map_functorObj 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (U : TopologicalSpace.Opens ↑X) : (TopologicalSpace.Opens.map f).obj (hf.functorObj U) = U
• Topology.IsInducing.functor_obj 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (U : TopologicalSpace.Opens ↑X) : hf.functor.obj U = hf.functorObj U
• Topology.IsInducing.opensGI 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) : GaloisInsertion (TopologicalSpace.Opens.map f).obj hf.functorObj
• Topology.IsInducing.le_functorObj_iff 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) {U : TopologicalSpace.Opens ↑X} {V : TopologicalSpace.Opens ↑Y} : V ≤ hf.functorObj U ↔ (TopologicalSpace.Opens.map f).obj V ≤ U
• Topology.IsInducing.mem_functorObj_iff 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (U : TopologicalSpace.Opens ↑X) {x : ↑X} : (CategoryTheory.ConcreteCategory.hom f) x ∈ hf.functorObj U ↔ x ∈ U
• Topology.IsInducing.functor_map 📋 Mathlib.Topology.Category.TopCat.Opens
  {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) {U V : TopologicalSpace.Opens ↑X} (h : U ⟶ V) : hf.functor.map h = CategoryTheory.homOfLE ⋯
• Topology.IsInducing.functorNhds 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) : CategoryTheory.Functor (TopologicalSpace.OpenNhds x) (TopologicalSpace.OpenNhds ((CategoryTheory.ConcreteCategory.hom f) x))
• Topology.IsInducing.adjunctionNhds 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) : TopologicalSpace.OpenNhds.map f x ⊣ h.functorNhds x
• Topology.IsInducing.functorNhds_obj_coe 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) (U : TopologicalSpace.OpenNhds x) : ↑((h.functorNhds x).obj U) = h.functor.obj ↑U
• Topology.IsInducing.functorNhds_map 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) {X✝ Y✝ : TopologicalSpace.OpenNhds x} (a✝ : ↑X✝ ⟶ ↑Y✝) : (h.functorNhds x).map a✝ = h.functor.map a✝
• TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_isInducing 📋 Mathlib.Topology.Sheaves.Stalks
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : TopCat} {f : X ⟶ Y} (hf : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (F : TopCat.Presheaf C X) (x : ↑X) : CategoryTheory.IsIso (TopCat.Presheaf.stalkPushforward C f F x)
• WeakPseudoEMetricSpace.IsInducing 📋 Mathlib.Topology.EMetricSpace.Defs
  {α : Type u_2} {β : Type u_3} [e : TopologicalSpace α] [n : TopologicalSpace β] {f : α → β} (hf : Topology.IsInducing f) (m : WeakPseudoEMetricSpace β) : WeakPseudoEMetricSpace α
• Topology.IsInducing.pseudoMetrizableSpace 📋 Mathlib.Topology.Metrizable.Basic
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace.PseudoMetrizableSpace Y] {f : X → Y} (hf : Topology.IsInducing f) : TopologicalSpace.PseudoMetrizableSpace X
• Topology.IsInducing.isSeparable_preimage 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {β : Type v} {α : Type u_2} [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] {f : β → α} [TopologicalSpace β] (hf : Topology.IsInducing f) {s : Set α} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable (f ⁻¹' s)
• Topology.IsInducing.comapPseudoMetricSpace 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [m : PseudoMetricSpace β] {f : α → β} (hf : Topology.IsInducing f) : PseudoMetricSpace α
• Topology.IsInducing.hasProd_iff 📋 Mathlib.Topology.Algebra.InfiniteSum.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {L : SummationFilter β} [CommMonoid γ] [TopologicalSpace γ] {G : Type u_4} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Topology.IsInducing ⇑g) (f : β → α) (a : α) : HasProd (⇑g ∘ f) (g a) L ↔ HasProd f a L
• Topology.IsInducing.hasSum_iff 📋 Mathlib.Topology.Algebra.InfiniteSum.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {L : SummationFilter β} [AddCommMonoid γ] [TopologicalSpace γ] {G : Type u_4} [FunLike G α γ] [AddMonoidHomClass G α γ] {g : G} (hg : Topology.IsInducing ⇑g) (f : β → α) (a : α) : HasSum (⇑g ∘ f) (g a) L ↔ HasSum f a L
• Topology.IsInducing.multipliable_iff_tprod_comp_mem_range 📋 Mathlib.Topology.Algebra.InfiniteSum.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {L : SummationFilter β} [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] {G : Type u_4} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Topology.IsInducing ⇑g) (f : β → α) : Multipliable f L ↔ Multipliable (⇑g ∘ f) L ∧ ∏'[L] (i : β), g (f i) ∈ Set.range ⇑g
• Topology.IsInducing.summable_iff_tsum_comp_mem_range 📋 Mathlib.Topology.Algebra.InfiniteSum.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {L : SummationFilter β} [AddCommMonoid γ] [TopologicalSpace γ] [T2Space γ] {G : Type u_4} [FunLike G α γ] [AddMonoidHomClass G α γ] {g : G} (hg : Topology.IsInducing ⇑g) (f : β → α) : Summable f L ↔ Summable (⇑g ∘ f) L ∧ ∑'[L] (i : β), g (f i) ∈ Set.range ⇑g
• AntilipschitzWith.isInducing 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : Continuous f) : Topology.IsInducing f
• Topology.IsInducing.frechetUrysohnSpace 📋 Mathlib.Topology.Sequences
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [FrechetUrysohnSpace Y] {f : X → Y} (hf : Topology.IsInducing f) : FrechetUrysohnSpace X
• Topology.IsInducing.perfectlyNormalSpace 📋 Mathlib.Topology.Separation.GDelta
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [PerfectlyNormalSpace Y] {e : X → Y} (he : Topology.IsInducing e) : PerfectlyNormalSpace X
• MeasurableEmbedding.borelSpace 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [TopologicalSpace α] [MeasurableSpace β] [TopologicalSpace β] [hβ : BorelSpace β] {e : α → β} (h'e : MeasurableEmbedding e) (h''e : Topology.IsInducing e) : BorelSpace α
• ContinuousMap.isInducing_postcomp 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Topology.IsInducing ⇑g) : Topology.IsInducing g.comp
• Topology.IsInducing.isPathConnected_iff 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {F : Set X} {f : X → Y} (hf : Topology.IsInducing f) : IsPathConnected F ↔ IsPathConnected (f '' F)
• Topology.IsInducing.joinedIn_image 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {F : Set X} {f : X → Y} (hf : Topology.IsInducing f) (hx : x ∈ F) (hy : y ∈ F) : JoinedIn (f '' F) (f x) (f y) ↔ JoinedIn F x y
• Topology.IsInducing.alexandrovDiscrete 📋 Mathlib.Topology.AlexandrovDiscrete
  {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [AlexandrovDiscrete α] {f : β → α} (h : Topology.IsInducing f) : AlexandrovDiscrete β
• Topology.IsInducing.locallyConvexSpace 📋 Mathlib.Topology.Algebra.Module.LocallyConvex
  {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] [LocallyConvexSpace 𝕜 F] [TopologicalSpace E] {f : E →ₗ[𝕜] F} (hf : Topology.IsInducing ⇑f) : LocallyConvexSpace 𝕜 E
• Topology.IsInducing.polynormableSpace 📋 Mathlib.Analysis.LocallyConvex.WithSeminorms
  {𝕜 : Type u_2} {𝕜₂ : Type u_3} {E : Type u_6} {F : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [TopologicalSpace F] [PolynormableSpace 𝕜₂ F] [TopologicalSpace E] {f : E →ₛₗ[σ₁₂] F} (hf : Topology.IsInducing ⇑f) : PolynormableSpace 𝕜 E
• Topology.IsInducing.withSeminorms 📋 Mathlib.Analysis.LocallyConvex.WithSeminorms
  {𝕜 : Type u_2} {𝕜₂ : Type u_3} {E : Type u_6} {F : Type u_7} {ι : Type u_9} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [TopologicalSpace F] {q : SeminormFamily 𝕜₂ F ι} (hq : WithSeminorms q) [TopologicalSpace E] {f : E →ₛₗ[σ₁₂] F} (hf : Topology.IsInducing ⇑f) : WithSeminorms (q.comp f)
• Seminorm.exists_le_comp_of_isInducing 📋 Mathlib.Analysis.LocallyConvex.WithSeminorms
  {𝕜₂ : Type u_3} {E : Type u_6} {F : Type u_7} [AddCommGroup E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace F] {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] [Module 𝕜 E] [TopologicalSpace E] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {p : Seminorm 𝕜 E} (hp : Continuous ⇑p) [PolynormableSpace 𝕜₂ F] {f : E →ₛₗ[σ₁₂] F} (hf : Topology.IsInducing ⇑f) : ∃ p₂, Continuous ⇑p₂ ∧ p ≤ p₂.comp f
• Seminorm.bound_comp_of_isInducing 📋 Mathlib.Analysis.LocallyConvex.WithSeminorms
  {𝕜₂ : Type u_3} {E : Type u_6} {F : Type u_7} {ι : Type u_9} [AddCommGroup E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace F] {𝕜 : Type u_11} [NontriviallyNormedField 𝕜] [Module 𝕜 E] [TopologicalSpace E] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {p : Seminorm 𝕜 E} (hp : Continuous ⇑p) {q : SeminormFamily 𝕜₂ F ι} (hq : WithSeminorms q) {f : E →ₛₗ[σ₁₂] F} (hf : Topology.IsInducing ⇑f) : ∃ s C, C ≠ 0 ∧ p ≤ (C • s.sup q).comp f
• UniformFun.continuousSMul_induced_of_range_bounded 📋 Mathlib.Topology.Algebra.Module.UniformConvergence
  (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) (H : Type u_4) {hom : Type u_5} [NormedField 𝕜] [AddCommGroup H] [Module 𝕜 H] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace H] [UniformSpace E] [IsUniformAddGroup E] [ContinuousSMul 𝕜 E] [FunLike hom H (α → E)] [LinearMapClass hom 𝕜 H (α → E)] (φ : hom) (hφ : Topology.IsInducing (⇑UniformFun.ofFun ∘ ⇑φ)) (h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))) : ContinuousSMul 𝕜 H
• UniformOnFun.continuousSMul_induced_of_image_bounded 📋 Mathlib.Topology.Algebra.Module.UniformConvergence
  (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) (H : Type u_4) {hom : Type u_5} [NormedField 𝕜] [AddCommGroup H] [Module 𝕜 H] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace H] [UniformSpace E] [IsUniformAddGroup E] [ContinuousSMul 𝕜 E] {𝔖 : Set (Set α)} [FunLike hom H (α → E)] [LinearMapClass hom 𝕜 H (α → E)] (φ : hom) (hφ : Topology.IsInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ ⇑φ)) (h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)) : ContinuousSMul 𝕜 H
• ContinuousLinearMap.isInducing_postcomp 📋 Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap
  {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedField 𝕜₃] {σ : 𝕜₁ →+* 𝕜₂} {τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] {E : Type u_4} {F : Type u_5} {G : Type u_6} [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [AddCommGroup G] [Module 𝕜₃ G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [IsTopologicalAddGroup F] [IsTopologicalAddGroup G] (f : F →SL[τ] G) (hf : Topology.IsInducing ⇑f) : Topology.IsInducing f.comp
• Topology.IsInducing.matrix_map 📋 Mathlib.Topology.Instances.Matrix
  {m : Type u_11} {n : Type u_12} {R : Type u_13} {S : Type u_14} [TopologicalSpace R] [TopologicalSpace S] {f : R → S} (hf : Topology.IsInducing f) : Topology.IsInducing fun x => x.map f
• BoundedContinuousFunction.isInducing_coeFn 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] : Topology.IsInducing (⇑UniformFun.ofFun ∘ DFunLike.coe)
• AlgebraicGeometry.isAffineHom_of_isInducing 📋 Mathlib.AlgebraicGeometry.Morphisms.Affine
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (hf₁ : Topology.IsInducing ⇑f) (hf₂ : IsClosed (Set.range ⇑f)) : AlgebraicGeometry.IsAffineHom f
• ContinuousLinearMap.isThetaTVS_comp 📋 Mathlib.Analysis.Asymptotics.TVS
  {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [NontriviallyNormedField 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [AddCommGroup F] [TopologicalSpace F] [Module 𝕜 F] {l : Filter α} {f : α → E} (g : E →L[𝕜] F) (hg : Topology.IsInducing ⇑g) : (⇑g ∘ f) =Θ[𝕜; l] f
• HasFDerivAt.isTheta_sub 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (hf : HasFDerivAt f f' x) (hf' : Topology.IsInducing ⇑f') : (fun x_1 => f x_1 - f x) =Θ[nhds x] fun x_1 => x_1 - x
• HasFDerivWithinAt.isTheta_sub 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hf' : Topology.IsInducing ⇑f') : (fun x_1 => f x_1 - f x) =Θ[nhdsWithin x s] fun x_1 => x_1 - x
• HasFDerivAtFilter.isTheta_sub 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {L : Filter (E × E)} (hf : HasFDerivAtFilter f f' L) (hf' : Topology.IsInducing ⇑f') : (fun p => f p.1 - f p.2) =Θ[L] fun p => p.1 - p.2

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c