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Result

Found 270 declarations mentioning Topology.IsEmbedding. Of these, only the first 200 are shown.

• Topology.IsEmbedding 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Prop
• Topology.IsEmbedding.injective 📋 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) : Function.Injective f
• Topology.IsClosedEmbedding.toIsEmbedding 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (self : Topology.IsClosedEmbedding f) : Topology.IsEmbedding f
• 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.IsOpenEmbedding.toIsEmbedding 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (self : Topology.IsOpenEmbedding f) : Topology.IsEmbedding f
• 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.IsClosedEmbedding.mk 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (toIsEmbedding : Topology.IsEmbedding f) (isClosed_range : IsClosed (Set.range f)) : Topology.IsClosedEmbedding f
• Topology.IsOpenEmbedding.mk 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : X → Y} (toIsEmbedding : Topology.IsEmbedding f) (isOpen_range : IsOpen (Set.range f)) : Topology.IsOpenEmbedding f
• Topology.isClosedEmbedding_iff 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Topology.IsClosedEmbedding f ↔ Topology.IsEmbedding f ∧ IsClosed (Set.range f)
• Topology.isOpenEmbedding_iff 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Topology.IsOpenEmbedding f ↔ Topology.IsEmbedding f ∧ IsOpen (Set.range f)
• Topology.IsEmbedding.id 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} [TopologicalSpace X] : Topology.IsEmbedding id
• Topology.IsEmbedding.of_subsingleton 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Subsingleton X] (f : X → Y) : Topology.IsEmbedding f
• Function.Injective.isEmbedding_induced 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [t : TopologicalSpace Y] (hf : Function.Injective f) : Topology.IsEmbedding f
• Topology.IsClosedEmbedding.isEmbedding 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsClosedEmbedding f) : Topology.IsEmbedding f
• Topology.IsEmbedding.continuous 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) : Continuous f
• Topology.IsEmbedding.discreteTopology 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] [DiscreteTopology Y] (hf : Topology.IsEmbedding f) : DiscreteTopology X
• Topology.IsEmbedding.induced 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [t : TopologicalSpace Y] (hf : Function.Injective f) : Topology.IsEmbedding 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.IsOpenEmbedding.isEmbedding 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsOpenEmbedding f) : Topology.IsEmbedding f
• Topology.IsEmbedding.isOpenEmbedding_of_surjective 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (hsurj : Function.Surjective f) : Topology.IsOpenEmbedding f
• Topology.IsOpenEmbedding.of_isEmbedding 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (hsurj : Function.Surjective f) : Topology.IsOpenEmbedding f
• Topology.IsClosedEmbedding.of_isEmbedding_isClosedMap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Topology.IsEmbedding f) (h₂ : IsClosedMap f) : Topology.IsClosedEmbedding f
• Topology.IsOpenEmbedding.of_isEmbedding_isOpenMap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Topology.IsEmbedding f) (h₂ : IsOpenMap f) : Topology.IsOpenEmbedding 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.isOpenEmbedding_iff_isEmbedding_isOpenMap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : Topology.IsOpenEmbedding f ↔ Topology.IsEmbedding f ∧ IsOpenMap f
• Function.LeftInverse.isEmbedding 📋 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 : Continuous f) (hg : Continuous g) : Topology.IsEmbedding g
• Topology.IsEmbedding.of_leftInverse 📋 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 : Continuous f) (hg : Continuous g) : Topology.IsEmbedding g
• Topology.IsEmbedding.closure_eq_preimage_closure_image 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (s : Set X) : closure s = f ⁻¹' closure (f '' s)
• Topology.IsEmbedding.map_nhds_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (x : X) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)
• Topology.IsEmbedding.mk' 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (inj : Function.Injective f) (induced : ∀ (x : X), Filter.comap f (nhds (f x)) = nhds x) : Topology.IsEmbedding f
• Topology.IsClosedEmbedding.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 X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : Topology.IsEmbedding g) (hgf : Topology.IsClosedEmbedding (g ∘ f)) : Topology.IsClosedEmbedding f
• Topology.IsEmbedding.comp 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : Topology.IsEmbedding g) (hf : Topology.IsEmbedding f) : Topology.IsEmbedding (g ∘ f)
• Topology.IsEmbedding.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 X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : Topology.IsEmbedding g) : Continuous f ↔ Continuous (g ∘ f)
• Topology.IsEmbedding.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 X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : Topology.IsEmbedding g) : Topology.IsEmbedding (g ∘ f) ↔ Topology.IsEmbedding f
• Topology.IsEmbedding.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 X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : Continuous f) (hg : Continuous g) (hgf : Topology.IsEmbedding (g ∘ f)) : Topology.IsEmbedding f
• Topology.IsEmbedding.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.IsEmbedding g) : Filter.Tendsto f l (nhds y) ↔ Filter.Tendsto (g ∘ f) l (nhds (g y))
• Topology.IsEmbedding.map_nhds_of_mem 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (x : X) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)
• 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
• Homeomorph.isEmbedding 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Topology.IsEmbedding ⇑h
• Homeomorph.comp_isEmbedding_iff 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : Y ≃ₜ Z) {f : X → Y} : Topology.IsEmbedding (⇑e ∘ f) ↔ Topology.IsEmbedding f
• Homeomorph.isEmbedding_comp_iff 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) {f : Y → Z} : Topology.IsEmbedding (f ∘ ⇑e) ↔ Topology.IsEmbedding f
• Topology.IsEmbedding.inl 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] : Topology.IsEmbedding Sum.inl
• Topology.IsEmbedding.inr 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] : Topology.IsEmbedding Sum.inr
• isEmbedding_prodMkRight 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X) : Topology.IsEmbedding (Prod.mk x)
• isEmbedding_prodMkLeft 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (y : Y) : Topology.IsEmbedding fun x => (x, y)
• isEmbedding_graph 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : Topology.IsEmbedding fun x => (x, f x)
• Topology.IsEmbedding.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.IsEmbedding (Sum.elim f g)) : Topology.IsEmbedding f
• Topology.IsEmbedding.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.IsEmbedding (Sum.elim f g)) : Topology.IsEmbedding g
• Topology.IsEmbedding.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.IsEmbedding f) (hg : Topology.IsEmbedding g) : Topology.IsEmbedding (Prod.map f g)
• Topology.IsEmbedding.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.IsEmbedding f) (hg : Topology.IsEmbedding g) (hsep : SeparatedNhds (Set.range f) (Set.range g)) : Topology.IsEmbedding (Sum.elim f g)
• Topology.IsEmbedding.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.IsEmbedding f) (hg : Topology.IsEmbedding g) (hFg : Disjoint (closure (Set.range f)) (Set.range g)) (hfG : Disjoint (Set.range f) (closure (Set.range g))) : Topology.IsEmbedding (Sum.elim f g)
• isEmbedding_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.IsEmbedding (Sum.elim f g) ↔ Topology.IsEmbedding f ∧ Topology.IsEmbedding g ∧ Disjoint (closure (Set.range f)) (Set.range g) ∧ Disjoint (Set.range f) (closure (Set.range g))
• Topology.IsEmbedding.uliftDown 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] : Topology.IsEmbedding ULift.down
• Topology.IsEmbedding.subtypeVal 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {p : X → Prop} : Topology.IsEmbedding Subtype.val
• Topology.IsEmbedding.sigmaMk 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {σ : ι → Type u_4} [(i : ι) → TopologicalSpace (σ i)] {i : ι} : Topology.IsEmbedding (Sigma.mk i)
• Function.Surjective.isEmbedding_comp 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {n : Type u_6} {m : Type u_7} (f : m → n) (hf : Function.Surjective f) : Topology.IsEmbedding fun x => x ∘ f
• Topology.IsEmbedding.codRestrict 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {e : X → Y} (he : Topology.IsEmbedding e) (s : Set Y) (hs : ∀ (x : X), e x ∈ s) : Topology.IsEmbedding (Set.codRestrict e s hs)
• Topology.IsEmbedding.inclusion 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {s t : Set X} (h : s ⊆ t) : Topology.IsEmbedding (Set.inclusion h)
• Topology.IsEmbedding.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.IsEmbedding (f i)) : Topology.IsEmbedding (Pi.map f)
• Topology.IsEmbedding.restrict 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s : Set X} {t : Set Y} (H : Set.MapsTo f s t) : Topology.IsEmbedding (Set.MapsTo.restrict f s t H)
• Topology.isEmbedding_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)} (h : Function.Injective f₁) : Topology.IsEmbedding (Sigma.map f₁ f₂) ↔ ∀ (i : ι), Topology.IsEmbedding (f₂ i)
• Topology.IsEmbedding.map_nhdsWithin_eq 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsEmbedding f) (s : Set α) (x : α) : Filter.map f (nhdsWithin x s) = nhdsWithin (f x) (f '' s)
• Topology.IsEmbedding.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.IsEmbedding g) {s : Set α} : ContinuousOn f s ↔ ContinuousOn (g ∘ f) s
• Topology.IsEmbedding.firstCountableTopology 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] {β : Type u_1} [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Topology.IsEmbedding f) : FirstCountableTopology α
• Topology.IsEmbedding.secondCountableTopology 📋 Mathlib.Topology.Bases
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : α → β} [TopologicalSpace β] [SecondCountableTopology β] (hf : Topology.IsEmbedding f) : SecondCountableTopology α
• Topology.IsEmbedding.separableSpace 📋 Mathlib.Topology.Bases
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] {f : α → β} (hf : Topology.IsEmbedding f) : TopologicalSpace.SeparableSpace α
• Topology.IsEmbedding.isCompact_iff 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Topology.IsEmbedding f) : IsCompact s ↔ IsCompact (f '' s)
• Topology.IsEmbedding.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.IsEmbedding f) : IsSigmaCompact s ↔ IsSigmaCompact (f '' s)
• Topology.IsEmbedding.t0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space Y] {f : X → Y} (hf : Topology.IsEmbedding f) : T0Space X
• Topology.IsEmbedding.t1Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {f : X → Y} (hf : Topology.IsEmbedding f) : T1Space X
• 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
• Topology.IsEmbedding.image_mem_codiscreteWithin_range 📋 Mathlib.Topology.DiscreteSubset
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s : Set X} : f '' s ∈ Filter.codiscreteWithin (Set.range f) ↔ s ∈ Filter.codiscrete X
• Topology.IsEmbedding.image_mem_codiscreteWithin 📋 Mathlib.Topology.DiscreteSubset
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s t : Set X} : f '' s ∈ Filter.codiscreteWithin (f '' t) ↔ s ∈ Filter.codiscreteWithin t
• Topology.IsEmbedding.t2Space 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Topology.IsEmbedding f) : T2Space X
• Topology.IsEmbedding.isLindelof_iff 📋 Mathlib.Topology.Compactness.Lindelof
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Topology.IsEmbedding f) : IsLindelof s ↔ IsLindelof (f '' s)
• Topology.IsEmbedding.t25Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T25Space Y] {f : X → Y} (hf : Topology.IsEmbedding f) : T25Space X
• Topology.IsEmbedding.t3Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {f : X → Y} (hf : Topology.IsEmbedding f) : T3Space X
• Topology.IsEmbedding.t5Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T5Space Y] {e : X → Y} (he : Topology.IsEmbedding e) : T5Space X
• IsDenseEmbedding.isEmbedding 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : IsDenseEmbedding e) : Topology.IsEmbedding e
• Topology.IsEmbedding.isTotallyDisconnected_range 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Topology.IsEmbedding f) : IsTotallyDisconnected (Set.range f) ↔ TotallyDisconnectedSpace α
• Topology.IsEmbedding.isTotallyDisconnected 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : Topology.IsEmbedding f) (h : IsTotallyDisconnected (f '' s)) : IsTotallyDisconnected s
• Topology.IsEmbedding.isTotallyDisconnected_image 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : Topology.IsEmbedding f) : IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s
• IsHomeomorph.isEmbedding 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsHomeomorph f) : Topology.IsEmbedding f
• Topology.IsEmbedding.toHomeomorphOfSurjective 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (hsurj : Function.Surjective f) : X ≃ₜ Y
• isHomeomorph_iff_isEmbedding_surjective 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : IsHomeomorph f ↔ Topology.IsEmbedding f ∧ Function.Surjective f
• Topology.IsEmbedding.uliftMap 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) : Topology.IsEmbedding (ULift.map f)
• Topology.IsEmbedding.toHomeomorph 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) : X ≃ₜ ↑(Set.range f)
• Topology.IsEmbedding.toHomeomorphOfSurjective_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (hsurj : Function.Surjective f) (a✝ : X) : (hf.toHomeomorphOfSurjective hsurj) a✝ = f a✝
• Topology.IsEmbedding.homeomorphImage 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (s : Set X) : ↑s ≃ₜ ↑(f '' s)
• Topology.IsEmbedding.homeomorphOfSubsetRange 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f) : ↑(f ⁻¹' s) ≃ₜ ↑s
• Topology.IsEmbedding.toHomeomorph_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (a : X) : ↑(hf.toHomeomorph a) = f a
• Topology.IsEmbedding.toHomeomorph_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (x : X) : hf.toHomeomorph.symm ⟨f x, ⋯⟩ = x
• Topology.IsEmbedding.homeomorphOfSubsetRange_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f) (x : ↑(f ⁻¹' s)) : ↑((hf.homeomorphOfSubsetRange hs) x) = f ↑x
• AddUnits.isEmbedding_val_mk' 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_4} [AddMonoid M] [TopologicalSpace M] {f : M → M} (hc : ContinuousOn f {x | IsAddUnit x}) (hf : ∀ (u : AddUnits M), f ↑u = ↑(-u)) : Topology.IsEmbedding AddUnits.val
• Units.isEmbedding_val_mk' 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_4} [Monoid M] [TopologicalSpace M] {f : M → M} (hc : ContinuousOn f {x | IsUnit x}) (hf : ∀ (u : Mˣ), f ↑u = ↑u⁻¹) : Topology.IsEmbedding Units.val
• AddUnits.embedding_val_mk 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_4} [SubtractionMonoid M] [TopologicalSpace M] (h : ContinuousOn Neg.neg {x | IsAddUnit x}) : Topology.IsEmbedding AddUnits.val
• Units.embedding_val_mk 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_4} [DivisionMonoid M] [TopologicalSpace M] (h : ContinuousOn Inv.inv {x | IsUnit x}) : Topology.IsEmbedding Units.val
• AddUnits.isEmbedding_embedProduct 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [AddMonoid M] : Topology.IsEmbedding ⇑(AddUnits.embedProduct M)
• Units.isEmbedding_embedProduct 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [Monoid M] : Topology.IsEmbedding ⇑(Units.embedProduct M)
• Topology.IsEmbedding.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.IsEmbedding ⇑f) : Topology.IsEmbedding ⇑(AddUnits.map f)
• Topology.IsEmbedding.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.IsEmbedding ⇑f) : Topology.IsEmbedding ⇑(Units.map f)
• Topology.IsEmbedding.comapUniformSpace 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [u : UniformSpace β] (f : α → β) (h : Topology.IsEmbedding f) : UniformSpace α
• IsUniformEmbedding.isEmbedding 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : IsUniformEmbedding f) : Topology.IsEmbedding f
• Embedding.to_isUniformEmbedding 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [u : UniformSpace β] (f : α → β) (h : Topology.IsEmbedding f) : IsUniformEmbedding f
• UniformOnFun.isEmbedding_toFun_finite 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  (α : Type u_1) (β : Type u_2) [UniformSpace β] : Topology.IsEmbedding ⇑(UniformOnFun.toFun {s | s.Finite})
• AddMonoidHom.isUniformEmbedding_of_isEmbedding 📋 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.IsEmbedding ⇑f) : IsUniformEmbedding ⇑f
• MonoidHom.isUniformEmbedding_of_isEmbedding 📋 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.IsEmbedding ⇑f) : IsUniformEmbedding ⇑f
• Units.isEmbedding_val₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
  {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] : Topology.IsEmbedding Units.val
• StrictMono.isEmbedding_of_ordConnected 📋 Mathlib.Topology.Order.Basic
  {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : (Set.range f).OrdConnected) : Topology.IsEmbedding f
• OrderEmbedding.isEmbedding_of_ordConnected 📋 Mathlib.Topology.Order.Basic
  {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [OrderTopology β] (f : α ↪o β) (hc : (Set.range ⇑f).OrdConnected) : Topology.IsEmbedding ⇑f
• Topology.IsEmbedding.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.IsEmbedding f) : IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ s' ∩ Set.range f = f '' s
• Set.restrictPreimage_isEmbedding 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Topology.IsEmbedding f) : Topology.IsEmbedding (s.restrictPreimage f)
• Topology.IsEmbedding.restrictPreimage 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Topology.IsEmbedding f) : Topology.IsEmbedding (s.restrictPreimage f)
• TopologicalSpace.IsOpenCover.isEmbedding_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.IsEmbedding f ↔ ∀ (i : ι), Topology.IsEmbedding ((U i).carrier.restrictPreimage f)
• isEmbedding_of_iSup_eq_top_of_preimage_subset_range 📋 Mathlib.Topology.LocalAtTarget
  {X : Type u_6} {Y : Type u_7} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Continuous f) {ι : Type u_4} (U : ι → TopologicalSpace.Opens Y) (hU : Set.range f ⊆ ↑(iSup U)) (V : ι → Type u_5) [(i : ι) → TopologicalSpace (V i)] (iV : (i : ι) → V i → X) (hiV : ∀ (i : ι), Continuous (iV i)) (hV : ∀ (i : ι), f ⁻¹' ↑(U i) ⊆ Set.range (iV i)) (hV' : ∀ (i : ι), Topology.IsEmbedding (f ∘ iV i)) : Topology.IsEmbedding f
• IsQuasiSeparated.image_of_isEmbedding 📋 Mathlib.Topology.QuasiSeparated
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (H : IsQuasiSeparated s) (h : Topology.IsEmbedding f) : IsQuasiSeparated (f '' s)
• IsRetrocompact.image_of_isEmbedding 📋 Mathlib.Topology.Constructible
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hs : IsRetrocompact s) (hfemb : Topology.IsEmbedding f) (hfcomp : IsRetrocompact (Set.range f)) : IsRetrocompact (f '' s)
• PrimeSpectrum.localization_comap_isEmbedding 📋 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.IsEmbedding (PrimeSpectrum.comap (algebraMap R S))
• PrimeSpectrum.isEmbedding_comap_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.IsEmbedding (PrimeSpectrum.comap f)
• TopCat.isEmbedding_iff 📋 Mathlib.Topology.Category.TopCat.Basic
  ⦃A X : TopCat⦄ (f : A ⟶ X) : TopCat.isEmbedding f ↔ Topology.IsEmbedding ⇑(TopCat.Hom.hom f)
• TopCat.isEmbedding_prodMap 📋 Mathlib.Topology.Category.TopCat.Limits.Products
  {W X Y Z : TopCat} {f : W ⟶ X} {g : Y ⟶ Z} (hf : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) (hg : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom g)) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.prod.map f g))
• TopCat.pullbackHomeoPreimage 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Z) (hf : Continuous f) (g : Y → Z) (hg : Topology.IsEmbedding g) : { p // f p.1 = g p.2 } ≃ₜ ↑(f ⁻¹' Set.range g)
• TopCat.fst_iso_of_right_embedding_range_subset 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S} (hg : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom g)) (H : Set.range ⇑(CategoryTheory.ConcreteCategory.hom f) ⊆ Set.range ⇑(CategoryTheory.ConcreteCategory.hom g)) : CategoryTheory.IsIso (CategoryTheory.Limits.pullback.fst f g)
• TopCat.snd_iso_of_left_embedding_range_subset 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y S : TopCat} {f : X ⟶ S} (hf : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) (g : Y ⟶ S) (H : Set.range ⇑(CategoryTheory.ConcreteCategory.hom g) ⊆ Set.range ⇑(CategoryTheory.ConcreteCategory.hom f)) : CategoryTheory.IsIso (CategoryTheory.Limits.pullback.snd f g)
• TopCat.fst_isEmbedding_of_right 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S} (H : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom g)) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.pullback.fst f g))
• TopCat.snd_isEmbedding_of_left 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y S : TopCat} {f : X ⟶ S} (H : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) (g : Y ⟶ S) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.pullback.snd f g))
• TopCat.isEmbedding_pullback_to_prod 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.pullback.fst f g) (CategoryTheory.Limits.pullback.snd f g)))
• TopCat.isEmbedding_of_pullback 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y S : TopCat} {f : X ⟶ S} {g : Y ⟶ S} (H₁ : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) (H₂ : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom g)) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.limit.π (CategoryTheory.Limits.cospan f g) CategoryTheory.Limits.WalkingCospan.one))
• TopCat.pullback_map_isEmbedding 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom i₁)) (H₂ : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom i₂)) (i₃ : S ⟶ T) (eq₁ : CategoryTheory.CategoryStruct.comp f₁ i₃ = CategoryTheory.CategoryStruct.comp i₁ g₁) (eq₂ : CategoryTheory.CategoryStruct.comp f₂ i₃ = CategoryTheory.CategoryStruct.comp i₂ g₂) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂))
• Submodule.isEmbedding_subtype 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Topology.IsEmbedding ⇑p.subtype
• Submodule.isEmbedding_subtypeL 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Topology.IsEmbedding ⇑p.subtypeL
• CommRingCat.HomTopology.isEmbedding_hom 📋 Mathlib.Algebra.Category.Ring.Topology
  (R A : CommRingCat) [TopologicalSpace ↑R] : Topology.IsEmbedding fun f => ⇑(CommRingCat.Hom.hom f)
• CommRingCat.HomTopology.isEmbedding_precomp_of_surjective 📋 Mathlib.Algebra.Category.Ring.Topology
  {R A B : CommRingCat} [TopologicalSpace ↑R] (f : A ⟶ B) (hf : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom f)) : Topology.IsEmbedding fun x => CategoryTheory.CategoryStruct.comp f x
• CommRingCat.HomTopology.isEmbedding_pushout 📋 Mathlib.Algebra.Category.Ring.Topology
  {R A B C : CommRingCat} [TopologicalSpace ↑R] [IsTopologicalRing ↑R] (φ : A ⟶ B) (ψ : A ⟶ C) : Topology.IsEmbedding fun f => (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl φ ψ) f, CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inr φ ψ) f)
• PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks 📋 Mathlib.RingTheory.Spectrum.Prime.TensorProduct
  (R : Type u_1) (S : Type u_2) (T : Type u_3) [CommRing R] [CommRing S] [Algebra R S] [CommRing T] [Algebra R T] (hRT : (algebraMap R T).SurjectiveOnStalks) : Topology.IsEmbedding (PrimeSpectrum.tensorProductTo R S T)
• Topology.IsEmbedding.metrizableSpace 📋 Mathlib.Topology.Metrizable.Basic
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace.MetrizableSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) : TopologicalSpace.MetrizableSpace X
• Topology.IsEmbedding.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.IsEmbedding f) {s : Set α} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable (f ⁻¹' s)
• NNReal.isEmbedding_coe 📋 Mathlib.Topology.MetricSpace.Basic
  : Topology.IsEmbedding NNReal.toReal
• Topology.IsEmbedding.comapMetricSpace 📋 Mathlib.Topology.MetricSpace.Basic
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [m : MetricSpace β] (f : α → β) (h : Topology.IsEmbedding f) : MetricSpace α
• ENNReal.isEmbedding_coe 📋 Mathlib.Topology.Algebra.Ring.Real
  : Topology.IsEmbedding ENNReal.ofNNReal
• Rat.isEmbedding_coe_real 📋 Mathlib.Topology.Instances.Rat
  : Topology.IsEmbedding Rat.cast
• AntilipschitzWith.isEmbedding 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : Continuous f) : Topology.IsEmbedding f
• Isometry.isEmbedding 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : Topology.IsEmbedding f
• Topology.IsEmbedding.to_isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [PseudoMetricSpace β] {f : α → β} (h : Topology.IsEmbedding f) : Isometry f
• Dilation.isEmbedding 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_1} {β : Type u_2} {F : Type u_4} [EMetricSpace α] [FunLike F α β] [PseudoEMetricSpace β] [DilationClass F α β] (f : F) : Topology.IsEmbedding ⇑f
• EReal.isEmbedding_coe_ennreal 📋 Mathlib.Topology.Instances.EReal.Lemmas
  : Topology.IsEmbedding ENNReal.toEReal
• EReal.isEmbedding_coe 📋 Mathlib.Topology.Instances.EReal.Lemmas
  : Topology.IsEmbedding Real.toEReal
• Topology.IsEmbedding.t6Space 📋 Mathlib.Topology.Separation.GDelta
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T6Space Y] {e : X → Y} (he : Topology.IsEmbedding e) : T6Space X
• Topology.IsEmbedding.measurableEmbedding 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [mα : MeasurableSpace α] [BorelSpace α] [mβ : TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h₁ : Topology.IsEmbedding f) (h₂ : MeasurableSet (Set.range f)) : MeasurableEmbedding f
• Embedding.comp_stronglyMeasurable_iff 📋 Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Topology.IsEmbedding g) : (MeasureTheory.StronglyMeasurable fun x => g (f x)) ↔ MeasureTheory.StronglyMeasurable f
• Topology.IsEmbedding.aestronglyMeasurable_comp_iff 📋 Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β] [TopologicalSpace γ] {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace.PseudoMetrizableSpace β] [TopologicalSpace.PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Topology.IsEmbedding g) : MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
• Metric.PiNatEmbed.exists_embedding_to_hilbert_cube 📋 Mathlib.Topology.MetricSpace.PiNat
  {X : Type u_3} [MetricSpace X] [TopologicalSpace.SeparableSpace X] : ∃ F, Topology.IsEmbedding F
• LinearIsometry.isEmbedding 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_5} {F : Type u_7} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f : F →ₛₗᵢ[σ₁₂] E₂) : Topology.IsEmbedding ⇑f
• Topology.IsEmbedding.toPartialHomeomorph 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] : PartialHomeomorph X Y
• Topology.IsEmbedding.toPartialHomeomorph_apply 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] : ↑(Topology.IsEmbedding.toPartialHomeomorph f h) = f
• PartialHomeomorph.isEmbedding 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (h : e.source = Set.univ) : Topology.IsEmbedding ↑e
• Topology.IsEmbedding.toPartialHomeomorph_source 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] : (Topology.IsEmbedding.toPartialHomeomorph f h).source = Set.univ
• Topology.IsEmbedding.toPartialHomeomorph_target 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] : (Topology.IsEmbedding.toPartialHomeomorph f h).target = Set.range f
• Topology.IsEmbedding.toPartialHomeomorph_left_inv 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] {x : X} : ↑(Topology.IsEmbedding.toPartialHomeomorph f h).symm (f x) = x
• Topology.IsEmbedding.toPartialHomeomorph_right_inv 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Topology.IsEmbedding f) [Nonempty X] {x : Y} (hx : x ∈ Set.range f) : f (↑(Topology.IsEmbedding.toPartialHomeomorph f h).symm x) = x
• PartialHomeomorph.isEmbedding_restrict 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) : Topology.IsEmbedding (e.source.domRestrict ↑e.toPartialEquiv)
• ContinuousMap.isEmbedding_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.IsEmbedding ⇑g) : Topology.IsEmbedding g.comp
• AffineMap.isEmbedding_linear_iff 📋 Mathlib.Topology.Algebra.Affine
  {R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : Topology.IsEmbedding ⇑f.linear ↔ Topology.IsEmbedding ⇑f
• SeparationQuotient.isEmbedding_outCLM 📋 Mathlib.Topology.Algebra.SeparationQuotient.Section
  (K : Type u_1) (E : Type u_2) [DivisionRing K] [AddCommGroup E] [Module K E] [TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul K E] : Topology.IsEmbedding ⇑(SeparationQuotient.outCLM K E)
• UniformConvergenceCLM.isEmbedding_coeFn 📋 Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM
  {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) : Topology.IsEmbedding (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe)
• ContinuousLinearMap.isEmbedding_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.IsEmbedding ⇑f) : Topology.IsEmbedding f.comp
• ContinuousLinearMap.isEmbedding_restrictScalars 📋 Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] {F : Type u_3} [AddCommGroup F] [TopologicalSpace F] [IsTopologicalAddGroup F] [Module 𝕜 F] (𝕜' : Type u_4) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [Module 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F] : Topology.IsEmbedding (ContinuousLinearMap.restrictScalars 𝕜')
• ContinuousMultilinearMap.isEmbedding_toUniformOnFun 📋 Mathlib.Topology.Algebra.Module.Multilinear.Topology
  {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [IsUniformAddGroup F] : Topology.IsEmbedding ContinuousMultilinearMap.toUniformOnFun
• ContinuousMultilinearMap.isEmbedding_restrictScalars 📋 Mathlib.Topology.Algebra.Module.Multilinear.Topology
  {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [IsTopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] {𝕜' : Type u_5} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [(i : ι) → Module 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F] : Topology.IsEmbedding (ContinuousMultilinearMap.restrictScalars 𝕜')
• ContinuousLinearMap.antilipschitz_of_isEmbedding 📋 Mathlib.Analysis.Normed.Operator.NormedSpace
  {𝕜 : Type u_1} {E : Type u_5} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) (hf : Topology.IsEmbedding ⇑f) : ∃ K, AntilipschitzWith K ⇑f
• Topology.IsEmbedding.isStrictMap 📋 Mathlib.Topology.Maps.Strict.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (f_emb : Topology.IsEmbedding f) : Topology.IsStrictMap f
• Topology.isEmbedding_iff_isStrictMap_injective 📋 Mathlib.Topology.Maps.Strict.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : Topology.IsEmbedding f ↔ Topology.IsStrictMap f ∧ Function.Injective f
• Topology.isStrictMap_iff_isEmbedding_kerLift 📋 Mathlib.Topology.Maps.Strict.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : Topology.IsStrictMap f ↔ Topology.IsEmbedding (Setoid.kerLift f)
• Topology.IsEmbedding.isStrictMap_iff 📋 Mathlib.Topology.Maps.Strict.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (g_emb : Topology.IsEmbedding g) : Topology.IsStrictMap f ↔ Topology.IsStrictMap (g ∘ f)
• Topology.IsEmbedding.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.IsEmbedding f) : Topology.IsEmbedding fun x => x.map f
• BoundedContinuousFunction.isEmbedding_coeFn 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] : Topology.IsEmbedding (⇑UniformFun.ofFun ∘ DFunLike.coe)
• AlgebraicGeometry.isEmbedding_isZariskiLocalAtTarget 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  : AlgebraicGeometry.IsZariskiLocalAtTarget (AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => Topology.IsEmbedding)
• AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsEmbedding 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  : (AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => Topology.IsEmbedding).RespectsIso
• AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback 📋 Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks
  {X Y S : AlgebraicGeometry.Scheme} (f : X ⟶ S) (g : Y ⟶ S) [AlgebraicGeometry.SurjectiveOnStalks g] : Topology.IsEmbedding fun x => ((CategoryTheory.Limits.pullback.fst f g) x, (CategoryTheory.Limits.pullback.snd f g) x)
• AlgebraicGeometry.isPreimmersion_eq_inf 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  : @AlgebraicGeometry.IsPreimmersion = @AlgebraicGeometry.SurjectiveOnStalks ⊓ AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => Topology.IsEmbedding
• AlgebraicGeometry.IsPreimmersion.mk_SpecMap 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {R S : CommRingCat} {f : R ⟶ S} (h₁ : Topology.IsEmbedding (PrimeSpectrum.comap (CommRingCat.Hom.hom f))) (h₂ : (CommRingCat.Hom.hom f).SurjectiveOnStalks) : AlgebraicGeometry.IsPreimmersion (AlgebraicGeometry.Spec.map f)
• AlgebraicGeometry.IsPreimmersion.SpecMap_iff 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {R S : CommRingCat} (f : R ⟶ S) : AlgebraicGeometry.IsPreimmersion (AlgebraicGeometry.Spec.map f) ↔ Topology.IsEmbedding (PrimeSpectrum.comap (CommRingCat.Hom.hom f)) ∧ (CommRingCat.Hom.hom f).SurjectiveOnStalks
• AlgebraicGeometry.IsPreimmersion.isEmbedding 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [self : AlgebraicGeometry.IsPreimmersion f] : Topology.IsEmbedding ⇑f
• AlgebraicGeometry.Scheme.Hom.isEmbedding 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [self : AlgebraicGeometry.IsPreimmersion f] : Topology.IsEmbedding ⇑f
• AlgebraicGeometry.IsPreimmersion.mk 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [toSurjectiveOnStalks : AlgebraicGeometry.SurjectiveOnStalks f] (isEmbedding : Topology.IsEmbedding ⇑f) : AlgebraicGeometry.IsPreimmersion f
• AlgebraicGeometry.isPreimmersion_iff 📋 Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsPreimmersion f ↔ AlgebraicGeometry.SurjectiveOnStalks f ∧ Topology.IsEmbedding ⇑f
• ENat.isEmbedding_natCast 📋 Mathlib.Topology.Instances.ENat
  : Topology.IsEmbedding Nat.cast
• ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap 📋 Mathlib.Topology.Algebra.Module.Alternating.Topology
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {ι : Type u_4} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [IsTopologicalAddGroup F] : Topology.IsEmbedding ContinuousAlternatingMap.toContinuousMultilinearMap
• ContinuousAlternatingMap.isEmbedding_restrictScalars 📋 Mathlib.Topology.Algebra.Module.Alternating.Topology
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {ι : Type u_4} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 E] {𝕜' : Type u_5} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [Module 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F] : Topology.IsEmbedding (ContinuousAlternatingMap.restrictScalars 𝕜')
• ContinuousMapZero.isEmbedding_toContinuousMap 📋 Mathlib.Topology.ContinuousMap.ContinuousMapZero
  {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] : Topology.IsEmbedding toContinuousMap
• HasFDerivAt.of_comp_of_isEmbedding 📋 Mathlib.Analysis.Calculus.FDeriv.OfCompLeft
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : E → F} {f : F → G} {h : E → G} {g' : E →L[𝕜] F} {f' : F →L[𝕜] G} {a : E} (hg : ContinuousAt g a) (hf : HasFDerivAt f f' (g a)) (hf' : Topology.IsEmbedding ⇑f') (hh : HasFDerivAt h (f' ∘SL g') a) (hcomp : f ∘ g =ᶠ[nhds a] h) : HasFDerivAt g g' a
• HasStrictFDerivAt.of_comp_of_isEmbedding 📋 Mathlib.Analysis.Calculus.FDeriv.OfCompLeft
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : E → F} {f : F → G} {h : E → G} {g' : E →L[𝕜] F} {f' : F →L[𝕜] G} {a : E} (hg : ContinuousAt g a) (hf : HasStrictFDerivAt f f' (g a)) (hf' : Topology.IsEmbedding ⇑f') (hh : HasStrictFDerivAt h (f' ∘SL g') a) (hcomp : f ∘ g =ᶠ[nhds a] h) : HasStrictFDerivAt g g' a
• HasFDerivAtFilter.of_comp_of_isEmbedding 📋 Mathlib.Analysis.Calculus.FDeriv.OfCompLeft
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : E → F} {f : F → G} {h : E → G} {g' : E →L[𝕜] F} {f' : F →L[𝕜] G} {lE : Filter (E × E)} {lF : Filter (F × F)} (hg : Filter.Tendsto (Prod.map g g) lE lF) (hf : HasFDerivAtFilter f f' lF) (hf' : Topology.IsEmbedding ⇑f') (hh : HasFDerivAtFilter h (f' ∘SL g') lE) (hcomp : Prod.map (f ∘ g) (f ∘ g) =ᶠ[lE] Prod.map h h) : HasFDerivAtFilter g g' lE
• HasFDerivWithinAt.of_comp_of_isEmbedding 📋 Mathlib.Analysis.Calculus.FDeriv.OfCompLeft
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : E → F} {f : F → G} {h : E → G} {g' : E →L[𝕜] F} {f' : F →L[𝕜] G} {a : E} {s : Set E} {t : Set F} (hg : Filter.Tendsto g (nhdsWithin a s) (nhdsWithin (g a) t)) (hf : HasFDerivWithinAt f f' t (g a)) (hf' : Topology.IsEmbedding ⇑f') (hh : HasFDerivWithinAt h (f' ∘SL g') s a) (hcomp : f ∘ g =ᶠ[nhdsWithin a s] h) (ha : a ∈ s) : HasFDerivWithinAt g g' s a
• TopPair.of 📋 Mathlib.Topology.Category.TopPair
  {A X : TopCat} (f : A ⟶ X) (h : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) : TopPair
• TopPair.isEmbedding_map 📋 Mathlib.Topology.Category.TopPair
  (X : TopPair) : Topology.IsEmbedding ⇑(CategoryTheory.ConcreteCategory.hom TopPair.map)
• Topology.IsEmbedding.isSimplyConnected_image 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) {s : Set X} : IsSimplyConnected (f '' s) ↔ IsSimplyConnected s

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 69fae59