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Result

Found 113 declarations mentioning TopologicalSpace.induced.

• TopologicalSpace.induced 📋 Mathlib.Topology.Defs.Induced
  {X : Type u_1} {Y : Type u_2} (f : X → Y) (t : TopologicalSpace Y) : TopologicalSpace X
• 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
• induced_fun_id 📋 Mathlib.Topology.Order
  {α : Type u_1} {t : TopologicalSpace α} : TopologicalSpace.induced (fun x => x) t = t
• induced_id 📋 Mathlib.Topology.Order
  {α : Type u_1} [t : TopologicalSpace α] : TopologicalSpace.induced id t = t
• continuous_induced_dom 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace β} : Continuous f
• WithTopology.topology_eq_induced 📋 Mathlib.Topology.Order
  {X : Type u_4} (t : TopologicalSpace X) : WithTopology.instTopologicalSpace X t = TopologicalSpace.induced WithTopology.ofTopology t
• isClosed_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set β} (h : IsClosed s) : IsClosed (f ⁻¹' s)
• isOpen_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set β} (h : IsOpen s) : IsOpen (f ⁻¹' s)
• gc_coinduced_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} (f : α → β) : GaloisConnection (TopologicalSpace.coinduced f) (TopologicalSpace.induced f)
• nhds_induced 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [T : TopologicalSpace α] (f : β → α) (a : β) : nhds a = Filter.comap f (nhds (f a))
• induced_generateFrom_eq 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {b : Set (Set β)} {f : α → β} : TopologicalSpace.induced f (TopologicalSpace.generateFrom b) = TopologicalSpace.generateFrom (Set.preimage f '' b)
• induced_compose 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {tγ : TopologicalSpace γ} {f : α → β} {g : β → γ} : TopologicalSpace.induced f (TopologicalSpace.induced g tγ) = TopologicalSpace.induced (g ∘ f) tγ
• map_nhds_induced_eq 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} (a : α) : Filter.map f (nhds a) = nhdsWithin (f a) (Set.range f)
• map_nhds_induced_of_surjective 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [T : TopologicalSpace α] {f : β → α} (hf : Function.Surjective f) (a : β) : Filter.map f (nhds a) = nhds (f a)
• Continuous.le_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {t : TopologicalSpace α} {t' : TopologicalSpace β} {f : α → β} (h : Continuous f) : t ≤ TopologicalSpace.induced f t'
• continuous_iff_le_induced 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β} : Continuous f ↔ t₁ ≤ TopologicalSpace.induced f t₂
• continuous_induced_rng 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} {γ : Type u_1} {f : α → β} {g : γ → α} {t₂ : TopologicalSpace β} {t₁ : TopologicalSpace γ} : Continuous g ↔ Continuous (f ∘ g)
• induced_iff_nhds_eq 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : β → α) : tβ = TopologicalSpace.induced f tα ↔ ∀ (b : β), nhds b = Filter.comap f (nhds (f b))
• isClosed_induced_iff 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {s : Set α} {f : α → β} : IsClosed s ↔ ∃ t_1, IsClosed t_1 ∧ f ⁻¹' t_1 = s
• isOpen_induced_iff 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {s : Set α} {f : α → β} : IsOpen s ↔ ∃ t_1, IsOpen t_1 ∧ f ⁻¹' t_1 = s
• closure_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {a : α} {s : Set α} : a ∈ closure s ↔ f a ∈ closure (f '' s)
• induced_mono 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {t₁ t₂ : TopologicalSpace α} {g : β → α} (h : t₁ ≤ t₂) : TopologicalSpace.induced g t₁ ≤ TopologicalSpace.induced g t₂
• isClosed_induced_iff' 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set α} : IsClosed s ↔ ∀ (a : α), f a ∈ closure (f '' s) → a ∈ s
• coinduced_le_iff_le_induced 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {f : α → β} {tα : TopologicalSpace α} {tβ : TopologicalSpace β} : TopologicalSpace.coinduced f tα ≤ tβ ↔ tα ≤ TopologicalSpace.induced f tβ
• map_nhds_induced_of_mem 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {a : α} (h : Set.range f ∈ nhds (f a)) : Filter.map f (nhds a) = nhds (f a)
• isOpen_induced_eq 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set α} : IsOpen s ↔ s ∈ Set.preimage f '' {s | IsOpen s}
• le_induced_generateFrom 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] {b : Set (Set β)} {f : α → β} (h : ∀ a ∈ b, IsOpen (f ⁻¹' a)) : t ≤ TopologicalSpace.induced f (TopologicalSpace.generateFrom b)
• Equiv.coinduced_symm 📋 Mathlib.Topology.Order
  {α : Type u_4} {β : Type u_5} (e : α ≃ β) : TopologicalSpace.coinduced ⇑e.symm = TopologicalSpace.induced ⇑e
• Equiv.induced_symm 📋 Mathlib.Topology.Order
  {α : Type u_4} {β : Type u_5} (e : α ≃ β) : TopologicalSpace.induced ⇑e.symm = TopologicalSpace.coinduced ⇑e
• mem_nhds_induced 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [T : TopologicalSpace α] (f : β → α) (a : β) (s : Set β) : s ∈ nhds a ↔ ∃ u ∈ nhds (f a), f ⁻¹' u ⊆ s
• induced_sInf 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {g : β → α} {s : Set (TopologicalSpace α)} : TopologicalSpace.induced g (sInf s) = sInf (TopologicalSpace.induced g '' s)
• induced_iInf 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {g : β → α} {ι : Sort w} {t : ι → TopologicalSpace α} : TopologicalSpace.induced g (⨅ i, t i) = ⨅ i, TopologicalSpace.induced g (t i)
• induced_inf 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {t₁ t₂ : TopologicalSpace α} {g : β → α} : TopologicalSpace.induced g (t₁ ⊓ t₂) = TopologicalSpace.induced g t₁ ⊓ TopologicalSpace.induced g t₂
• induced_const 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] {x : α} : TopologicalSpace.induced (fun x_1 => x) t = ⊤
• induced_top 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {g : β → α} : TopologicalSpace.induced g ⊤ = ⊤
• Topology.IsInducing.induced 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] (f : X → Y) : Topology.IsInducing 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.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
• 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
• Homeomorph.induced_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : TopologicalSpace.induced (⇑h) inst✝ = inst✝¹
• prod_induced_induced 📋 Mathlib.Topology.Constructions.SumProd
  {Y : Type v} {W : Type u_1} [TopologicalSpace Y] [TopologicalSpace W] {X : Type u_5} {Z : Type u_6} (f : X → Y) (g : Z → W) : instTopologicalSpaceProd = TopologicalSpace.induced (fun p => (f p.1, g p.2)) instTopologicalSpaceProd
• 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
• induced_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 : X → (i : ι) → A i) : TopologicalSpace.induced f Pi.topologicalSpace = ⨅ i, TopologicalSpace.induced (fun x => f x i) inferInstance
• Pi.induced_precomp 📋 Mathlib.Topology.Constructions
  {Y : Type v} {ι : Type u_2} [TopologicalSpace Y] {ι' : Type u_6} (φ : ι' → ι) : TopologicalSpace.induced (fun x => x ∘ φ) Pi.topologicalSpace = ⨅ i', TopologicalSpace.induced (Function.eval (φ i')) inst✝
• Pi.induced_precomp' 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] {ι' : Type u_6} (φ : ι' → ι) : TopologicalSpace.induced (fun f j => f (φ j)) Pi.topologicalSpace = ⨅ i', TopologicalSpace.induced (Function.eval (φ i')) (T (φ i'))
• Pi.induced_domRestrict 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (S : Set ι) : TopologicalSpace.induced S.domRestrict Pi.topologicalSpace = ⨅ i ∈ S, TopologicalSpace.induced (Function.eval i) (T i)
• Pi.induced_restrict 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (S : Set ι) : TopologicalSpace.induced S.domRestrict Pi.topologicalSpace = ⨅ i ∈ S, TopologicalSpace.induced (Function.eval i) (T i)
• Pi.induced_domRestrict_sUnion 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (𝔖 : Set (Set ι)) : TopologicalSpace.induced (⋃₀ 𝔖).domRestrict Pi.topologicalSpace = ⨅ S ∈ 𝔖, TopologicalSpace.induced S.domRestrict Pi.topologicalSpace
• Pi.induced_restrict_sUnion 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (𝔖 : Set (Set ι)) : TopologicalSpace.induced (⋃₀ 𝔖).domRestrict Pi.topologicalSpace = ⨅ S ∈ 𝔖, TopologicalSpace.induced S.domRestrict Pi.topologicalSpace
• nhdsSet_induced 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_3} {β : Type u_4} {t : TopologicalSpace β} (f : α → β) (s : Set α) : nhdsSet s = Filter.comap f (nhdsSet (f '' s))
• map_nhdsSet_induced_eq 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_3} {β : Type u_4} {t : TopologicalSpace β} {f : α → β} (s : Set α) : Filter.map f (nhdsSet s) = nhdsSetWithin (f '' s) (Set.range f)
• mem_nhdsSet_induced 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_3} {β : Type u_4} {t : TopologicalSpace β} (f : α → β) (s u : Set α) : u ∈ nhdsSet s ↔ ∃ v ∈ nhdsSet (f '' s), f ⁻¹' v ⊆ u
• TopologicalSpace.firstCountableTopology_induced 📋 Mathlib.Topology.Bases
  (α : Type u_1) (β : Type u_2) [t : TopologicalSpace β] [FirstCountableTopology β] (f : α → β) : FirstCountableTopology α
• TopologicalSpace.secondCountableTopology_induced 📋 Mathlib.Topology.Bases
  (α : Type u_1) (β : Type u_2) [t : TopologicalSpace β] [SecondCountableTopology β] (f : α → β) : SecondCountableTopology α
• TopologicalSpace.IsTopologicalBasis.induced 📋 Mathlib.Topology.Bases
  {β : Type u_1} {α : Type u_2} [s : TopologicalSpace β] (f : α → β) {T : Set (Set β)} (h : TopologicalSpace.IsTopologicalBasis T) : TopologicalSpace.IsTopologicalBasis (Set.preimage f '' T)
• TopologicalSpace.IsTopologicalBasis.inf_induced 📋 Mathlib.Topology.Bases
  {α : Type u} {β : Type u_1} [t : TopologicalSpace α] {γ : Type u_2} [s : TopologicalSpace β] {B₁ : Set (Set α)} {B₂ : Set (Set β)} (h₁ : TopologicalSpace.IsTopologicalBasis B₁) (h₂ : TopologicalSpace.IsTopologicalBasis B₂) (f₁ : γ → α) (f₂ : γ → β) : TopologicalSpace.IsTopologicalBasis (Set.image2 (fun x1 x2 => f₁ ⁻¹' x1 ∩ f₂ ⁻¹' x2) B₁ B₂)
• IsTopologicalBasis.iInf_induced 📋 Mathlib.Topology.Bases
  {β : Type u_1} {ι : Type u_2} {X : ι → Type u_3} [t : (i : ι) → TopologicalSpace (X i)] {T : (i : ι) → Set (Set (X i))} (cond : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (T i)) (f : (i : ι) → β → X i) : TopologicalSpace.IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, f i ⁻¹' U i}
• R1Space.induced 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R1Space X] (f : Y → X) : R1Space Y
• regularSpace_induced 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [RegularSpace X] (f : Y → X) : RegularSpace Y
• AddUnits.topology_eq_inf 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [AddMonoid M] : AddUnits.instTopologicalSpaceAddUnits = TopologicalSpace.induced AddUnits.val inst✝ ⊓ TopologicalSpace.induced (fun u => ↑(-u)) inst✝
• Units.topology_eq_inf 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] [Monoid M] : Units.instTopologicalSpaceUnits = TopologicalSpace.induced Units.val inst✝ ⊓ TopologicalSpace.induced (fun u => ↑u⁻¹) inst✝
• ContinuousSMul.induced 📋 Mathlib.Topology.Algebra.MulAction
  {R : Type u_5} {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β] (f : F) : ContinuousSMul R α
• continuousAdd_induced 📋 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 N] [ContinuousAdd N] (f : F) : ContinuousAdd M
• continuousMul_induced 📋 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 N] [ContinuousMul N] (f : F) : ContinuousMul M
• separatelyContinuousAdd_induced 📋 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 N] [SeparatelyContinuousAdd N] (f : F) : SeparatelyContinuousAdd M
• separatelyContinuousMul_induced 📋 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 N] [SeparatelyContinuousMul N] (f : F) : SeparatelyContinuousMul M
• ContinuousAdd.induced 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [Add α] [Add β] [AddHomClass F α β] [tβ : TopologicalSpace β] [ContinuousAdd β] (f : F) : ContinuousAdd α
• ContinuousMul.induced 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [Mul α] [Mul β] [MulHomClass F α β] [tβ : TopologicalSpace β] [ContinuousMul β] (f : F) : ContinuousMul α
• UniformSpace.toTopologicalSpace_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u : UniformSpace β} : (UniformSpace.comap f u).toTopologicalSpace = TopologicalSpace.induced f u.toTopologicalSpace
• UniformOnFun.topologicalSpace_eq 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  (α : Type u_1) (β : Type u_2) [UniformSpace β] (𝔖 : Set (Set α)) : UniformOnFun.topologicalSpace α β 𝔖 = ⨅ s ∈ 𝔖, TopologicalSpace.induced (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(UniformOnFun.toFun 𝔖)) (UniformFun.topologicalSpace (↑s) β)
• ContinuousInv.induced 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {α : Type u_4} {β : Type u_5} {F : Type u_6} [FunLike F α β] [Group α] [DivisionMonoid β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousInv β] (f : F) : ContinuousInv α
• ContinuousNeg.induced 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {α : Type u_4} {β : Type u_5} {F : Type u_6} [FunLike F α β] [AddGroup α] [SubtractionMonoid β] [AddMonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousNeg β] (f : F) : ContinuousNeg α
• isTopologicalAddGroup_induced 📋 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] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) : IsTopologicalAddGroup H
• isTopologicalGroup_induced 📋 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] [FunLike F H G] [MonoidHomClass F H G] (f : F) : IsTopologicalGroup H
• topologicalAddGroup_induced 📋 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] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) : IsTopologicalAddGroup H
• topologicalGroup_induced 📋 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] [FunLike F H G] [MonoidHomClass F H G] (f : F) : IsTopologicalGroup H
• induced_topology_le_preorder 📋 Mathlib.Topology.Order.Basic
  {α : Type u} {β : Type v} [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y : α}, f x < f y ↔ x < y) : TopologicalSpace.induced f inst✝ ≤ Preorder.topology α
• induced_orderTopology 📋 Mathlib.Topology.Order.Basic
  {α : Type u_1} {β : Type u_2} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H : ∀ {x y : β}, x < y → ∃ a, f a ∈ Set.Ioo x y) : OrderTopology α
• StrictMono.induced_topology_eq_preorder 📋 Mathlib.Topology.Order.Basic
  {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : (Set.range f).OrdConnected) : TopologicalSpace.induced f t = Preorder.topology α
• induced_orderTopology' 📋 Mathlib.Topology.Order.Basic
  {α : Type u_1} {β : Type u_2} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H₁ : ∀ {a : α} {x : β}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a : α} {x : β}, f a < x → ∃ b > a, f b ≤ x) : OrderTopology α
• induced_topology_eq_preorder 📋 Mathlib.Topology.Order.Basic
  {α : Type u} {β : Type v} [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y) (H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b) : TopologicalSpace.induced f inst✝ = Preorder.topology α
• TopCat.limit_topology 📋 Mathlib.Topology.Category.TopCat.Limits.Basic
  {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J TopCat) [CategoryTheory.Limits.HasLimit F] : (CategoryTheory.Limits.limit F).str = ⨅ j, TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.limit.π F j))) (F.obj j).str
• TopCat.induced_of_isLimit 📋 Mathlib.Topology.Category.TopCat.Limits.Basic
  {J : Type v} [CategoryTheory.Category.{w, v} J] {F : CategoryTheory.Functor J TopCat} (c : CategoryTheory.Limits.Cone F) (hc : CategoryTheory.Limits.IsLimit c) : c.pt.str = ⨅ j, TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom (c.π.app j))) (F.obj j).str
• TopCat.nonempty_isLimit_iff_eq_induced 📋 Mathlib.Topology.Category.TopCat.Limits.Basic
  {J : Type v} [CategoryTheory.Category.{w, v} J] {F : CategoryTheory.Functor J TopCat} (c : CategoryTheory.Limits.Cone F) (hc : CategoryTheory.Limits.IsLimit ((CategoryTheory.forget TopCat).mapCone c)) : Nonempty (CategoryTheory.Limits.IsLimit c) ↔ c.pt.str = ⨅ j, TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom (c.π.app j))) (F.obj j).str
• TopCat.prod_topology 📋 Mathlib.Topology.Category.TopCat.Limits.Products
  {X Y : TopCat} : (X ⨯ Y).str = TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom CategoryTheory.Limits.prod.fst)) X.str ⊓ TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom CategoryTheory.Limits.prod.snd)) Y.str
• TopCat.pullback_topology 📋 Mathlib.Topology.Category.TopCat.Limits.Pullbacks
  {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) : (CategoryTheory.Limits.pullback f g).str = TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.pullback.fst f g))) X.str ⊓ TopologicalSpace.induced (⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.pullback.snd f g))) Y.str
• continuousSMul_induced 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {M₁ : Type u_3} {M₂ : Type u_4} [SMul R M₁] [SMul R M₂] [u : TopologicalSpace R] {t : TopologicalSpace M₂} [ContinuousSMul R M₂] {F : Type u_6} [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] (f : F) : ContinuousSMul R M₁
• continuousSMul_inducedₛₗ 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {S : Type u_2} {M₁ : Type u_3} {M₂' : Type u_5} {φ : R → S} [SMul R M₁] [SMul S M₂'] [u : TopologicalSpace R] [u' : TopologicalSpace S] {t' : TopologicalSpace M₂'} [ContinuousSMul S M₂'] {F' : Type u_7} [FunLike F' M₁ M₂'] [MulActionSemiHomClass F' φ M₁ M₂'] (f' : F') (hφ : Continuous φ) : ContinuousSMul R M₁
• WeakEMetricSpace.induced 📋 Mathlib.Topology.EMetricSpace.Defs
  {α : Type u_3} {β : Type u_4} [n : TopologicalSpace β] {f : α → β} (hf : Function.Injective f) (m : WeakEMetricSpace β) : WeakEMetricSpace α
• borel_comap 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} {t : TopologicalSpace β} : borel α = MeasurableSpace.comap f (borel β)
• Equiv.polishSpace_induced 📋 Mathlib.Topology.MetricSpace.Polish
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] [PolishSpace β] (f : α ≃ β) : PolishSpace α
• ContinuousMap.compactOpen_le_induced 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X) : ContinuousMap.compactOpen ≤ TopologicalSpace.induced (ContinuousMap.restrict s) ContinuousMap.compactOpen
• ContinuousMap.compactOpen_eq_iInf_induced 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] : ContinuousMap.compactOpen = ⨅ K, ⨅ (_ : IsCompact K), TopologicalSpace.induced (ContinuousMap.restrict K) ContinuousMap.compactOpen
• LocallyConvexSpace.induced 📋 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] {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F] (f : E →ₗ[𝕜] F) : LocallyConvexSpace 𝕜 E
• PolynormableSpace.induced 📋 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] (f : E →ₛₗ[σ₁₂] F) : PolynormableSpace 𝕜 E
• LinearMap.withSeminorms_induced 📋 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) (f : E →ₛₗ[σ₁₂] F) : WithSeminorms (q.comp f)
• UniformOnFun.continuousSMul_submodule_of_image_bounded 📋 Mathlib.Topology.Algebra.Module.UniformConvergence
  (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [UniformSpace E] [IsUniformAddGroup E] [ContinuousSMul 𝕜 E] {𝔖 : Set (Set α)} (H : Submodule 𝕜 (UniformOnFun α E 𝔖)) (h : ∀ u ∈ H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (u '' s)) : ContinuousSMul 𝕜 ↥H
• UniformConvergenceCLM.topologicalSpace_eq 📋 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)) : UniformConvergenceCLM.instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) (UniformOnFun.topologicalSpace E F 𝔖)
• LinearMap.mem_span_iff_continuous_of_finite 📋 Mathlib.Analysis.LocallyConvex.WeakDual
  {ι : Type u_4} {𝕜 : Type u_5} {E : Type u_6} [Finite ι] [Field 𝕜] [t𝕜 : TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [T0Space 𝕜] {f : ι → E →ₗ[𝕜] 𝕜} (φ : E →ₗ[𝕜] 𝕜) : φ ∈ Submodule.span 𝕜 (Set.range f) ↔ Continuous ⇑φ
• LinearMap.mem_span_iff_continuous 📋 Mathlib.Analysis.LocallyConvex.WeakDual
  {ι : Type u_4} {𝕜 : Type u_5} {E : Type u_6} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E →ₗ[𝕜] 𝕜} (φ : E →ₗ[𝕜] 𝕜) : φ ∈ Submodule.span 𝕜 (Set.range f) ↔ Continuous ⇑φ
• pullbackTopology_def 📋 Mathlib.Topology.FiberBundle.Constructions
  {B : Type u_1} (F : Type u_2) (E : B → Type u_3) {B' : Type u_4} (f : B' → B) [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] : pullbackTopology F E f = TopologicalSpace.induced Bundle.TotalSpace.proj inst✝ ⊓ TopologicalSpace.induced (Bundle.Pullback.lift f) inst✝¹
• induced_topology_pure 📋 Mathlib.Topology.Compactification.StoneCech
  {α : Type u} : TopologicalSpace.induced pure Ultrafilter.topologicalSpace = ⊥
• completelyRegularSpace_induced 📋 Mathlib.Topology.Separation.CompletelyRegular
  {X : Type u_1} {Y : Type u_2} {t : TopologicalSpace Y} (ht : CompletelyRegularSpace Y) (f : X → Y) : CompletelyRegularSpace X
• RestrictedProduct.topologicalSpace_eq_of_principal 📋 Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace
  {ι : Type u_1} {R : ι → Type u_2} {A : (i : ι) → Set (R i)} [(i : ι) → TopologicalSpace (R i)] {S : Set ι} : RestrictedProduct.topologicalSpace R A (Filter.principal S) = TopologicalSpace.induced DFunLike.coe inferInstance
• RestrictedProduct.topologicalSpace_eq_of_bot 📋 Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace
  {ι : Type u_1} {R : ι → Type u_2} {A : (i : ι) → Set (R i)} [(i : ι) → TopologicalSpace (R i)] : RestrictedProduct.topologicalSpace R A ⊥ = TopologicalSpace.induced DFunLike.coe inferInstance
• RestrictedProduct.topologicalSpace_eq_of_top 📋 Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace
  {ι : Type u_1} {R : ι → Type u_2} {A : (i : ι) → Set (R i)} [(i : ι) → TopologicalSpace (R i)] : RestrictedProduct.topologicalSpace R A ⊤ = TopologicalSpace.induced DFunLike.coe inferInstance
• ContinuousSMul.topology_eq_of_induced_eq 📋 Mathlib.Topology.Algebra.Module.EmbeddingOfLocal
  (𝕜₁ : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜₁] [AddCommGroup E] [Module 𝕜₁ E] (t₁ t₂ : TopologicalSpace E) [IsTopologicalAddGroup E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜₁ E] [ContinuousSMul 𝕜₁ E] {V : Set E} (V_mem : V ∈ nhds 0) (H : TopologicalSpace.induced Subtype.val t₁ = TopologicalSpace.induced Subtype.val t₂) : t₁ = t₂
• LinearMap.IsWeak.eq_induced 📋 Mathlib.Topology.Algebra.Module.IsWeak
  {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} {inst✝ : CommSemiring 𝕜} {inst✝¹ : TopologicalSpace 𝕜} {inst✝² : AddCommMonoid E} {inst✝³ : Module 𝕜 E} {inst✝⁴ : AddCommMonoid F} {inst✝⁵ : Module 𝕜 F} {t : TopologicalSpace E} {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} [self : B.IsWeak] : t = TopologicalSpace.induced (fun x1 x2 => (B x1) x2) Pi.topologicalSpace
• LinearMap.IsWeak.mk 📋 Mathlib.Topology.Algebra.Module.IsWeak
  {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [t : TopologicalSpace E] {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (eq_induced : t = TopologicalSpace.induced (fun x1 x2 => (B x1) x2) Pi.topologicalSpace) : B.IsWeak
• LinearMap.isWeak_iff 📋 Mathlib.Topology.Algebra.Module.IsWeak
  {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [t : TopologicalSpace E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : B.IsWeak ↔ t = TopologicalSpace.induced (fun x1 x2 => (B x1) x2) Pi.topologicalSpace
• TopologicalSpace.eq_induced_by_maps_to_sierpinski 📋 Mathlib.Topology.ContinuousMap.T0Sierpinski
  (X : Type u_1) [t : TopologicalSpace X] : t = ⨅ u, TopologicalSpace.induced (fun x => x ∈ u) sierpinskiSpace

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