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Result

Found 251 declarations mentioning T0Space. Of these, only the first 200 are shown.

• T0Space 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : Prop
• specializationOrder 📋 Mathlib.Topology.Separation.Basic
  (X : Type u_3) [TopologicalSpace X] [T0Space X] : PartialOrder X
• T1Space.t0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T1Space X] : T0Space X
• SeparationQuotient.instT0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] : T0Space (SeparationQuotient X)
• instNontrivialTopologyOfT0SpaceOfNontrivial 📋 Mathlib.Topology.Separation.Basic
  (X : Type u_1) [TopologicalSpace X] [T0Space X] [Nontrivial X] : NontrivialTopology X
• instT1SpaceOfT0SpaceOfR0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] [R0Space X] : T1Space X
• nontrivial_iff_nontrivialTopology 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Nontrivial X ↔ NontrivialTopology X
• subsingleton_iff_indiscreteTopology 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Subsingleton X ↔ IndiscreteTopology X
• ULift.instT0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : T0Space (ULift.{u_3, u_1} X)
• nhds_injective 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Function.Injective nhds
• t0Space_iff_nhds_injective 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Function.Injective nhds
• t1Space_iff_t0Space_and_r0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] : T1Space X ↔ T0Space X ∧ R0Space X
• inseparable_eq_eq 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Inseparable = Eq
• Subtype.t0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {p : X → Prop} : T0Space (Subtype p)
• t0Space_iff_not_inseparable 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ¬Inseparable x y
• Homeomorph.t0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] (h : X ≃ₜ Y) : T0Space Y
• Inseparable.eq 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} (h : Inseparable x y) : x = y
• T0Space.mk 📋 Mathlib.Topology.Separation.Basic
  {X : Type u} [TopologicalSpace X] (t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y) : T0Space X
• T0Space.t0 📋 Mathlib.Topology.Separation.Basic
  {X : Type u} {inst✝ : TopologicalSpace X} [self : T0Space X] ⦃x y : X⦄ : Inseparable x y → x = y
• inseparable_iff_eq 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} : Inseparable x y ↔ x = y
• t0Space_iff_inseparable 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ ∀ (x y : X), Inseparable x y → x = y
• 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
• Pi.instT0Space 📋 Mathlib.Topology.Separation.Basic
  {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), T0Space (X i)] : T0Space ((i : ι) → X i)
• Prod.instT0Space 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] [T0Space Y] : T0Space (X × 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
• exists_open_singleton_of_finite 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] [Finite X] [Nonempty X] : ∃ x, IsOpen {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
• nhds_eq_nhds_iff 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {a b : X} : nhds a = nhds b ↔ a = b
• t0Space_of_injective_of_continuous 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X
• minimal_nonempty_closed_subsingleton 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsClosed s) (hmin : ∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) : s.Subsingleton
• minimal_nonempty_open_subsingleton 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsOpen s) (hmin : ∀ t ⊆ s, t.Nonempty → IsOpen t → t = s) : s.Subsingleton
• t0Space_iff_exists_isOpen_xor'_mem 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• t0Space_iff_exists_isOpen_xor_mem 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• exists_isOpen_xor'_mem 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} (h : x ≠ y) : ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• exists_isOpen_xor_mem 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} (h : x ≠ y) : ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• exists_isOpen_singleton_of_isOpen_finite 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hfin : s.Finite) (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen {x}
• IsClosed.exists_closed_singleton 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] [CompactSpace X] {S : Set X} (hS : IsClosed S) (hne : S.Nonempty) : ∃ x ∈ S, IsClosed {x}
• T0Space.of_open_cover 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] (h : ∀ (x : X), ∃ s, x ∈ s ∧ IsOpen s ∧ T0Space ↑s) : T0Space X
• t0Space_iff_or_notMem_closure 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun a b => a ∉ closure {b} ∨ b ∉ closure {a}
• minimal_nonempty_closed_eq_singleton 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsClosed s) (hne : s.Nonempty) (hmin : ∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x}
• minimal_nonempty_open_eq_singleton 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsOpen s) (hne : s.Nonempty) (hmin : ∀ t ⊆ s, t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x}
• TopologicalSpace.IsTopologicalBasis.eq_iff 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] {b : Set (Set X)} (hb : TopologicalSpace.IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, x ∈ s ↔ y ∈ s
• T0Space.of_cover 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] (h : ∀ (x y : X), Inseparable x y → ∃ s, x ∈ s ∧ y ∈ s ∧ T0Space ↑s) : T0Space X
• t1Space_TFAE 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of, ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ nhds x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ nhds x, y ∉ s, ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (nhds x) (pure y), ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (nhds y), ∀ ⦃x y : X⦄, x ⤳ y → x = y, T0Space X ∧ R0Space X].TFAE
• instT2SpaceOfR1SpaceOfT0Space 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] [R1Space X] [T0Space X] : T2Space X
• R1Space.t2Space_iff_t0Space 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] [R1Space X] : T2Space X ↔ T0Space X
• T3Space.toT0Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u} {inst✝ : TopologicalSpace X} [self : T3Space X] : T0Space X
• instT3Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} [TopologicalSpace X] [T0Space X] [RegularSpace X] : T3Space X
• T3Space.mk 📋 Mathlib.Topology.Separation.Regular
  {X : Type u} [TopologicalSpace X] [toT0Space : T0Space X] [toRegularSpace : RegularSpace X] : T3Space X
• RegularSpace.t3Space_iff_t0Space 📋 Mathlib.Topology.Separation.Regular
  {X : Type u_1} [TopologicalSpace X] [RegularSpace X] : T3Space X ↔ T0Space X
• SeparationQuotient.lift' 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space β] (f : α → β) : SeparationQuotient α → β
• t0Space_iff_ker_uniformity 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] : T0Space α ↔ (uniformity α).ker = Set.diagonal α
• SeparationQuotient.uniformContinuous_lift' 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space β] (f : α → β) : UniformContinuous (SeparationQuotient.lift' f)
• eq_of_clusterPt_uniformity 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] [T0Space α] {x y : α} (h : ClusterPt (x, y) (uniformity α)) : x = y
• SeparationQuotient.lift'_mk 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space β] {f : α → β} (h : UniformContinuous f) (a : α) : SeparationQuotient.lift' f (SeparationQuotient.mk a) = f a
• eq_of_uniformity 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u_1} [UniformSpace α] [T0Space α] {x y : α} (h : ∀ {V : Set (α × α)}, V ∈ uniformity α → (x, y) ∈ V) : x = y
• eq_of_uniformity_basis 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u_1} [UniformSpace α] [T0Space α] {ι : Sort u_2} {p : ι → Prop} {s : ι → Set (α × α)} (hs : (uniformity α).HasBasis p s) {x y : α} (h : ∀ {i : ι}, p i → (x, y) ∈ s i) : x = y
• t0Space_iff_uniformity 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] : T0Space α ↔ ∀ (x y : α), (∀ r ∈ uniformity α, (x, y) ∈ r) → x = y
• eq_of_forall_symmetric 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u_1} [UniformSpace α] [T0Space α] {x y : α} (h : ∀ {V : Set (α × α)}, V ∈ uniformity α → SetRel.IsSymm V → (x, y) ∈ V) : x = y
• t0Space_iff_uniformity' 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] : T0Space α ↔ Pairwise fun x y => ∃ r ∈ uniformity α, (x, y) ∉ r
• isClosed_of_spaced_out 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ uniformity α) {s : Set α} (hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s
• isClosed_range_of_spaced_out 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] {ι : Type u_1} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ uniformity α) {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (Set.range f)
• IsUniformInducing.injective 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β} (h : IsUniformInducing f) : Function.Injective f
• IsUniformInducing.isUniformEmbedding 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β} (hf : IsUniformInducing f) : IsUniformEmbedding f
• isUniformEmbedding_iff_isUniformInducing 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β} : IsUniformEmbedding f ↔ IsUniformInducing f
• uniformly_extend_of_ind 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [T0Space γ] (b : β) : ⋯.extend f (e b) = f b
• isClosedEmbedding_of_spaced_out 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {β : Type v} [UniformSpace β] {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] [T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ uniformity β) (hf : Pairwise fun x y => (f x, f y) ∉ s) : Topology.IsClosedEmbedding f
• Dense.extend_of_ind 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : ↑s → β} [T0Space β] (hs : Dense s) (hf : UniformContinuous f) (x : ↑s) : hs.extend f ↑x = f x
• uniformly_extend_unique 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} [T0Space γ] {g : α → γ} (hg : ∀ (b : β), g (e b) = f b) (hc : Continuous g) : ⋯.extend f = g
• IsComplete.isClosed 📋 Mathlib.Topology.UniformSpace.CompleteSeparated
  {α : Type u_1} [UniformSpace α] [T0Space α] {s : Set α} (h : IsComplete s) : IsClosed s
• IsUniformEmbedding.isClosedEmbedding 📋 Mathlib.Topology.UniformSpace.CompleteSeparated
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] [CompleteSpace α] [T0Space β] {f : α → β} (hf : IsUniformEmbedding f) : Topology.IsClosedEmbedding f
• IsDenseInducing.continuous_extend_of_cauchy 📋 Mathlib.Topology.UniformSpace.CompleteSeparated
  {α : Type u_1} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] {γ : Type u_4} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {e : α → β} {f : α → γ} (de : IsDenseInducing e) (h : ∀ (b : β), Cauchy (Filter.map f (Filter.comap e (nhds b)))) : Continuous (de.extend f)
• IsDenseInducing.extend_Z_bilin 📋 Mathlib.Topology.Algebra.IsUniformGroup.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {G : Type u_5} [TopologicalSpace α] [AddCommGroup α] [IsTopologicalAddGroup α] [TopologicalSpace β] [AddCommGroup β] [TopologicalSpace γ] [AddCommGroup γ] [IsTopologicalAddGroup γ] [TopologicalSpace δ] [AddCommGroup δ] [UniformSpace G] [AddCommGroup G] {e : β →+ α} (de : IsDenseInducing ⇑e) {f : δ →+ γ} (df : IsDenseInducing ⇑f) {φ : β →+ δ →+ G} (hφ : Continuous fun p => (φ p.1) p.2) [IsUniformAddGroup G] [T0Space G] [CompleteSpace G] : Continuous (⋯.extend fun p => (φ p.1) p.2)
• TopologicalSpace.Opens.comap_injective 📋 Mathlib.Topology.Sets.Opens
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [T0Space β] : Function.Injective TopologicalSpace.Opens.comap
• TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom 📋 Mathlib.Topology.Sets.Closeds
  {α : Type u_2} [TopologicalSpace α] [T0Space α] {s : TopologicalSpace.Closeds α} (hs : IsAtom s) : ∃ a, ↑s = {a}
• genericPoints.finite 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] (h : (irreducibleComponents α).Finite) : (genericPoints α).Finite
• genericPoints.component_injective 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] : Function.Injective genericPoints.component
• genericPoints.equiv 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] [QuasiSober α] : ↑(genericPoints α) ≃ ↑(irreducibleComponents α)
• IsGenericPoint.eq 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] {x y : α} {S : Set α} [T0Space α] (h : IsGenericPoint x S) (h' : IsGenericPoint y S) : x = y
• genericPoints_eq_singleton 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [QuasiSober α] [IrreducibleSpace α] [T0Space α] : genericPoints α = {genericPoint α}
• genericPoints.ofComponent_component 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] [QuasiSober α] (x : ↑(genericPoints α)) : genericPoints.ofComponent (genericPoints.component x) = x
• irreducibleSetEquivPoints 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [QuasiSober α] [T0Space α] : TopologicalSpace.IrreducibleCloseds α ≃o α
• genericPoints.equiv_apply 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] [QuasiSober α] (x : ↑(genericPoints α)) : genericPoints.equiv x = genericPoints.component x
• genericPoints.equiv_symm_apply 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [T0Space α] [QuasiSober α] (x : ↑(irreducibleComponents α)) : genericPoints.equiv.symm x = genericPoints.ofComponent x
• coe_irreducibleEquivPoints_symm_apply 📋 Mathlib.Topology.Sober
  {α : Type u_1} [TopologicalSpace α] [QuasiSober α] [T0Space α] (x : α) : ↑(irreducibleSetEquivPoints.symm x) = closure {x}
• Topology.IsOpenEmbedding.coheight_eq 📋 Mathlib.Topology.KrullDimension
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [QuasiSober Y] [T0Space Y] [QuasiSober X] [T0Space X] {x : X} (f : X → Y) (hf : Topology.IsOpenEmbedding f) : Order.coheight (f x) = Order.coheight x
• SpectralSpace.toT0Space 📋 Mathlib.Topology.Spectral.Basic
  {X : Type u_3} {inst✝ : TopologicalSpace X} [self : SpectralSpace X] : T0Space X
• SpectralSpace.mk 📋 Mathlib.Topology.Spectral.Basic
  {X : Type u_3} [TopologicalSpace X] [toT0Space : T0Space X] [toCompactSpace : CompactSpace X] [toQuasiSober : QuasiSober X] [toQuasiSeparatedSpace : QuasiSeparatedSpace X] [toPrespectralSpace : PrespectralSpace X] : SpectralSpace X
• PrimeSpectrum.instT0Space 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u} [CommSemiring R] : T0Space (PrimeSpectrum R)
• EMetric.nontrivial_iff_nontrivialTopology 📋 Mathlib.Topology.EMetricSpace.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Nontrivial X ↔ NontrivialTopology X
• EMetric.subsingleton_iff_indiscreteTopology 📋 Mathlib.Topology.EMetricSpace.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : Subsingleton X ↔ IndiscreteTopology X
• EMetricSpace.instT0Space 📋 Mathlib.Topology.EMetricSpace.Basic
  {γ : Type w} [EMetricSpace γ] : T0Space γ
• EMetricSpace.ofT0PseudoEMetricSpace 📋 Mathlib.Topology.EMetricSpace.Basic
  (α : Type u_2) [PseudoEMetricSpace α] [T0Space α] : EMetricSpace α
• TopologicalSpace.MetrizableSpace.toT0Space 📋 Mathlib.Topology.Metrizable.Basic
  {X : Type u_5} {t : TopologicalSpace X} [self : TopologicalSpace.MetrizableSpace X] : T0Space X
• TopologicalSpace.MetrizableSpace.mk 📋 Mathlib.Topology.Metrizable.Basic
  {X : Type u_5} [t : TopologicalSpace X] [toPseudoMetrizableSpace : TopologicalSpace.PseudoMetrizableSpace X] [toT0Space : T0Space X] : TopologicalSpace.MetrizableSpace X
• TopologicalSpace.PseudoMetrizableSpace.toMetrizableSpace 📋 Mathlib.Topology.Metrizable.Basic
  {X : Type u_2} [TopologicalSpace X] [T0Space X] [h : TopologicalSpace.PseudoMetrizableSpace X] : TopologicalSpace.MetrizableSpace X
• MetricSpace.instT0Space 📋 Mathlib.Topology.MetricSpace.Basic
  {γ : Type w} [MetricSpace γ] : T0Space γ
• MetricSpace.ofT0PseudoMetricSpace 📋 Mathlib.Topology.MetricSpace.Basic
  (α : Type u_2) [PseudoMetricSpace α] [T0Space α] : MetricSpace α
• Multipliable.tprod_prod 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommGroup α] [UniformSpace α] [IsUniformGroup α] [CompleteSpace α] [T0Space α] {f : β × γ → α} (h : Multipliable f) : ∏' (p : β × γ), f p = ∏' (b : β) (c : γ), f (b, c)
• Summable.tsum_prod 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommGroup α] [UniformSpace α] [IsUniformAddGroup α] [CompleteSpace α] [T0Space α] {f : β × γ → α} (h : Summable f) : ∑' (p : β × γ), f p = ∑' (b : β) (c : γ), f (b, c)
• Multipliable.tprod_comm 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommGroup α] [UniformSpace α] [IsUniformGroup α] [CompleteSpace α] [T0Space α] {f : β → γ → α} (h : Multipliable (Function.uncurry f)) : ∏' (c : γ) (b : β), f b c = ∏' (b : β) (c : γ), f b c
• Summable.tsum_comm 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommGroup α] [UniformSpace α] [IsUniformAddGroup α] [CompleteSpace α] [T0Space α] {f : β → γ → α} (h : Summable (Function.uncurry f)) : ∑' (c : γ) (b : β), f b c = ∑' (b : β) (c : γ), f b c
• Multipliable.tprod_sigma 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [IsUniformGroup α] [CompleteSpace α] [T0Space α] {γ : β → Type u_4} {f : (b : β) × γ b → α} (ha : Multipliable f) : ∏' (p : (b : β) × γ b), f p = ∏' (b : β) (c : γ b), f ⟨b, c⟩
• Summable.tsum_sigma 📋 Mathlib.Topology.Algebra.InfiniteSum.Constructions
  {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [IsUniformAddGroup α] [CompleteSpace α] [T0Space α] {γ : β → Type u_4} {f : (b : β) × γ b → α} (ha : Summable f) : ∑' (p : (b : β) × γ b), f p = ∑' (b : β) (c : γ b), f ⟨b, c⟩
• UniformSpace.metricSpace 📋 Mathlib.Topology.Metrizable.Uniformity
  (X : Type u_2) [UniformSpace X] [(uniformity X).IsCountablyGenerated] [T0Space X] : MetricSpace X
• UniformSpace.metrizableSpace 📋 Mathlib.Topology.Metrizable.Uniformity
  {X : Type u_1} [UniformSpace X] [(uniformity X).IsCountablyGenerated] [T0Space X] : TopologicalSpace.MetrizableSpace X
• AbstractCompletion.ofComplete 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type uα} [UniformSpace α] [T0Space α] [CompleteSpace α] : AbstractCompletion.{uα, uα} α
• AbstractCompletion.separation 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type u} [UniformSpace α] (self : AbstractCompletion.{v, u} α) : T0Space self.space
• AbstractCompletion.mk 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type u} [UniformSpace α] (space : Type v) (coe : α → space) (uniformStruct : UniformSpace space) (complete : CompleteSpace space) (separation : T0Space space) (isUniformInducing : IsUniformInducing coe) (dense : DenseRange coe) : AbstractCompletion.{v, u} α
• AbstractCompletion.extend_comp_coe 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type uα} [UniformSpace α] (pkg : AbstractCompletion.{vα, uα} α) {β : Type uβ} [UniformSpace β] [CompleteSpace β] [T0Space β] {f : pkg.space → β} (hf : UniformContinuous f) : pkg.extend (f ∘ pkg.coe) = f
• AbstractCompletion.extension₂_coe_coe 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type uα} [UniformSpace α] (pkg : AbstractCompletion.{vα, uα} α) {β : Type uβ} [UniformSpace β] (pkg' : AbstractCompletion.{vβ, uβ} β) {γ : Type uγ} [UniformSpace γ] [T0Space γ] {f : α → β → γ} (hf : UniformContinuous (Function.uncurry f)) (a : α) (b : β) : pkg.extend₂ pkg' f (pkg.coe a) (pkg'.coe b) = f a b
• AbstractCompletion.extend_unique 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type uα} [UniformSpace α] (pkg : AbstractCompletion.{vα, uα} α) {β : Type uβ} [UniformSpace β] {f : α → β} [CompleteSpace β] [T0Space β] (hf : UniformContinuous f) {g : pkg.space → β} (hg : UniformContinuous g) (h : ∀ (a : α), f a = g (pkg.coe a)) : pkg.extend f = g
• AbstractCompletion.extend_map 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type uα} [UniformSpace α] (pkg : AbstractCompletion.{vα, uα} α) {β : Type uβ} [UniformSpace β] (pkg' : AbstractCompletion.{vβ, uβ} β) {γ : Type uγ} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β} (hf : UniformContinuous f) (hg : UniformContinuous g) : pkg'.extend f ∘ pkg.map pkg' g = pkg.extend (f ∘ g)
• UniformSpace.Completion.t0Space 📋 Mathlib.Topology.UniformSpace.Completion
  (α : Type u_1) [UniformSpace α] : T0Space (UniformSpace.Completion α)
• UniformSpace.Completion.coe_injective 📋 Mathlib.Topology.UniformSpace.Completion
  (α : Type u_1) [UniformSpace α] [T0Space α] : Function.Injective UniformSpace.Completion.coe'
• UniformSpace.Completion.isUniformEmbedding_coe 📋 Mathlib.Topology.UniformSpace.Completion
  (α : Type u_1) [UniformSpace α] [T0Space α] : IsUniformEmbedding UniformSpace.Completion.coe'
• UniformSpace.Completion.UniformCompletion.completeEquivSelf 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] [CompleteSpace α] [T0Space α] : UniformSpace.Completion α ≃ᵤ α
• UniformSpace.Completion.isDenseEmbedding_coe 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] [T0Space α] : IsDenseEmbedding UniformSpace.Completion.coe'
• UniformSpace.Completion.coe_inj 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] [T0Space α] {a b : α} : ↑a = ↑b ↔ a = b
• CauchyFilter.extend_pureCauchy 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] [T0Space β] {f : α → β} (hf : UniformContinuous f) (a : α) : CauchyFilter.extend f (CauchyFilter.pureCauchy a) = f a
• UniformSpace.Completion.extension_coe 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [T0Space β] (hf : UniformContinuous f) (a : α) : UniformSpace.Completion.extension f ↑a = f a
• CauchyFilter.separated_pureCauchy_injective 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] [T0Space α] : Function.Injective fun a => SeparationQuotient.mk (CauchyFilter.pureCauchy a)
• UniformSpace.Completion.extension₂_coe_coe 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] {f : α → β → γ} [T0Space γ] (hf : UniformContinuous₂ f) (a : α) (b : β) : UniformSpace.Completion.extension₂ f ↑a ↑b = f a b
• UniformSpace.Completion.extension_comp_coe 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] [T0Space β] [CompleteSpace β] {f : UniformSpace.Completion α → β} (hf : UniformContinuous f) : UniformSpace.Completion.extension (f ∘ UniformSpace.Completion.coe') = f
• UniformSpace.Completion.extension_unique 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [T0Space β] [CompleteSpace β] (hf : UniformContinuous f) {g : UniformSpace.Completion α → β} (hg : UniformContinuous g) (h : ∀ (a : α), f a = g ↑a) : UniformSpace.Completion.extension f = g
• UniformSpace.Completion.extension_map 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformSpace.Completion.extension f ∘ UniformSpace.Completion.map g = UniformSpace.Completion.extension (f ∘ g)
• CauchyFilter.cauchyFilter_eq 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] [CompleteSpace α] [T0Space α] {f g : CauchyFilter α} : (↑f).lim = (↑g).lim ↔ Inseparable f g
• T6Space.toT0Space 📋 Mathlib.Topology.Separation.GDelta
  {X : Type u} {inst✝ : TopologicalSpace X} [self : T6Space X] : T0Space X
• T6Space.mk 📋 Mathlib.Topology.Separation.GDelta
  {X : Type u} [TopologicalSpace X] [toT0Space : T0Space X] [toPerfectlyNormalSpace : PerfectlyNormalSpace X] : T6Space X
• separatesPointsOfOpensMeasurableSpaceOfT0Space 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [T0Space α] : MeasurableSpace.SeparatesPoints α
• OpensMeasurableSpace.separatesPoints 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [T0Space α] : MeasurableSpace.SeparatesPoints α
• borel_eq_top_of_countable 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {α : Type u_6} [TopologicalSpace α] [T0Space α] [Countable α] : borel α = ⊤
• TopologicalSpace.IsCompletelyMetrizableSpace_of_isCompletelyPseudoMetrizableSpace 📋 Mathlib.Topology.Metrizable.CompletelyMetrizable
  {X : Type u_1} [TopologicalSpace X] [TopologicalSpace.IsCompletelyPseudoMetrizableSpace X] [T0Space X] : TopologicalSpace.IsCompletelyMetrizableSpace X
• TopologicalSpace.IsCompletelyMetrizableSpace.of_completeSpace_metrizable 📋 Mathlib.Topology.Metrizable.CompletelyMetrizable
  {X : Type u_1} [UniformSpace X] [CompleteSpace X] [(uniformity X).IsCountablyGenerated] [T0Space X] : TopologicalSpace.IsCompletelyMetrizableSpace X
• TopologicalSpace.Opens.CompleteCopy.instT0Space 📋 Mathlib.Topology.MetricSpace.Polish
  {α : Type u_1} [MetricSpace α] {s : TopologicalSpace.Opens α} : T0Space s.CompleteCopy
• instHasCountableSeparatingOnIsOpenOfT0SpaceOfSecondCountableTopologyElem 📋 Mathlib.Topology.Separation.CountableSeparatingOn
  {X : Type u_1} [TopologicalSpace X] {s : Set X} [T0Space ↑s] [SecondCountableTopology ↑s] : HasCountableSeparatingOn X IsOpen s
• Measurable.borelSpace_codomain 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [T0Space Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f) (hsurj : Function.Surjective f) : BorelSpace Y
• Continuous.map_borel_eq 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f) : MeasurableSpace.map f (borel X) = borel Y
• Quotient.borelSpace 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] {s : Setoid X} [T0Space (Quotient s)] [SecondCountableTopology (Quotient s)] : BorelSpace (Quotient s)
• Continuous.map_eq_borel 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f) : MeasurableSpace.map f inst✝ = borel Y
• Measurable.map_measurableSpace_eq_borel 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [T0Space Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f) (hsurj : Function.Surjective f) : MeasurableSpace.map f inst✝ = borel Y
• ContinuousMap.instT0Space 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [T0Space Y] : T0Space C(X, Y)
• IsTopologicalAddTorsor.t0Space 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [T0Space V] : T0Space P
• IsTopologicalTorsor.t0Space 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [Group V] [TopologicalSpace V] [Torsor V P] [TopologicalSpace P] [IsTopologicalTorsor P] [T0Space V] : T0Space P
• UniformSpace.Completion.coe_eq_zero_iff 📋 Mathlib.Topology.Algebra.GroupCompletion
  {α : Type u_3} [UniformSpace α] [Zero α] [T0Space α] {x : α} : ↑x = 0 ↔ x = 0
• AddMonoidHom.extension 📋 Mathlib.Topology.Algebra.GroupCompletion
  {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [CompleteSpace β] [T0Space β] (f : α →+ β) (hf : Continuous ⇑f) : UniformSpace.Completion α →+ β
• AddMonoidHom.continuous_extension 📋 Mathlib.Topology.Algebra.GroupCompletion
  {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [CompleteSpace β] [T0Space β] (f : α →+ β) (hf : Continuous ⇑f) : Continuous ⇑(f.extension hf)
• AddMonoidHom.extension_coe 📋 Mathlib.Topology.Algebra.GroupCompletion
  {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [CompleteSpace β] [T0Space β] (f : α →+ β) (hf : Continuous ⇑f) (a : α) : (f.extension hf) ↑a = f a
• UniformSpace.Completion.coe_eq_one_iff 📋 Mathlib.Topology.Algebra.UniformRing
  (α : Type u_1) [Ring α] [UniformSpace α] [T0Space α] {x : α} : ↑x = 1 ↔ x = 1
• UniformSpace.Completion.extensionHom 📋 Mathlib.Topology.Algebra.UniformRing
  {α : Type u_1} [Ring α] [UniformSpace α] [IsTopologicalRing α] [IsUniformAddGroup α] {β : Type u} [UniformSpace β] [Ring β] [IsUniformAddGroup β] [IsTopologicalRing β] (f : α →+* β) (hf : Continuous ⇑f) [CompleteSpace β] [T0Space β] : UniformSpace.Completion α →+* β
• UniformSpace.Completion.extensionHom_coe 📋 Mathlib.Topology.Algebra.UniformRing
  {α : Type u_1} [Ring α] [UniformSpace α] [IsTopologicalRing α] [IsUniformAddGroup α] {β : Type u} [UniformSpace β] [Ring β] [IsUniformAddGroup β] [IsTopologicalRing β] (f : α →+* β) (hf : Continuous ⇑f) [CompleteSpace β] [T0Space β] (a : α) : (UniformSpace.Completion.extensionHom f hf) ↑a = f a
• Isometry.completion_extension 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] [CompleteSpace β] [T0Space β] {f : α → β} (h : Isometry f) : Isometry (UniformSpace.Completion.extension f)
• Isometry.extensionHom 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] [CompleteSpace β] [T0Space β] {f : α →+* β} (h : Isometry ⇑f) : UniformSpace.Completion α →+* β
• Isometry.extensionHom_coe 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] [CompleteSpace β] [T0Space β] {f : α →+* β} (h : Isometry ⇑f) (x : α) : h.extensionHom ↑x = f x
• ContinuousLinearMap.extend 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] (e : E →L[𝕜] Eₗ) : Eₗ →SL[σ₁₂] F
• ContinuousLinearMap.extend_unique 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) (g : Eₗ →SL[σ₁₂] F) (H : g ∘SL e = f) : f.extend e = g
• ContinuousLinearMap.extend_zero 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) : ContinuousLinearMap.extend 0 e = 0
• ContinuousLinearMap.extend_eq 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) (x : E) : (f.extend e) (e x) = f x
• ContinuousLinearMap.fromCompletion 📋 Mathlib.Topology.Algebra.LinearMapCompletion
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {S : Type u_4} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [Semiring S] [Module S α] [UniformContinuousConstSMul S α] [Semiring R] [UniformSpace β] [AddCommGroup β] [IsUniformAddGroup β] [Module R β] [UniformContinuousConstSMul R β] {σ : S →+* R} [T0Space β] [CompleteSpace β] (f : α →SL[σ] β) : UniformSpace.Completion α →SL[σ] β
• ContinuousLinearMap.fromCompletion_apply_coe 📋 Mathlib.Topology.Algebra.LinearMapCompletion
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {S : Type u_4} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [Semiring S] [Module S α] [UniformContinuousConstSMul S α] [Semiring R] [UniformSpace β] [AddCommGroup β] [IsUniformAddGroup β] [Module R β] [UniformContinuousConstSMul R β] {σ : S →+* R} [T0Space β] [CompleteSpace β] (f : α →SL[σ] β) (e : α) : f.fromCompletion ↑e = f e
• ContinuousLinearMap.coe_fromCompletion 📋 Mathlib.Topology.Algebra.LinearMapCompletion
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {S : Type u_4} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [Semiring S] [Module S α] [UniformContinuousConstSMul S α] [Semiring R] [UniformSpace β] [AddCommGroup β] [IsUniformAddGroup β] [Module R β] [UniformContinuousConstSMul R β] {σ : S →+* R} [T0Space β] [CompleteSpace β] (f : α →SL[σ] β) : ⇑f.fromCompletion = UniformSpace.Completion.extension ⇑f
• ContinuousLinearMap.toAddMonoidHom_fromCompletion 📋 Mathlib.Topology.Algebra.LinearMapCompletion
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {S : Type u_4} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [Semiring S] [Module S α] [UniformContinuousConstSMul S α] [Semiring R] [UniformSpace β] [AddCommGroup β] [IsUniformAddGroup β] [Module R β] [UniformContinuousConstSMul R β] {σ : S →+* R} [T0Space β] [CompleteSpace β] (f : α →SL[σ] β) : (↑f.fromCompletion).toAddMonoidHom = (↑f).toAddMonoidHom.extension ⋯
• ContinuousLinearMap.fromCompletion_unique 📋 Mathlib.Topology.Algebra.LinearMapCompletion
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {S : Type u_4} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [Semiring S] [Module S α] [UniformContinuousConstSMul S α] [Semiring R] [UniformSpace β] [AddCommGroup β] [IsUniformAddGroup β] [Module R β] [UniformContinuousConstSMul R β] {σ : S →+* R} [T0Space β] [CompleteSpace β] (f : α →SL[σ] β) (g : UniformSpace.Completion α →SL[σ] β) (h : ∀ (e : α), f e = g ↑e) : f.fromCompletion = g
• WithLp.instProdT0Space 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (WithLp p (α × β))
• PiLp.instProdT0Space 📋 Mathlib.Analysis.Normed.Lp.PiLp
  (p : ENNReal) {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → TopologicalSpace (β i)] [∀ (i : ι), T0Space (β i)] : T0Space (PiLp p β)
• Nat.sumByResidueClasses 📋 Mathlib.Analysis.SumOverResidueClass
  {R : Type u_1} [AddCommGroup R] [UniformSpace R] [IsUniformAddGroup R] [CompleteSpace R] [T0Space R] {f : ℕ → R} (hf : Summable f) (N : ℕ) [NeZero N] : ∑' (n : ℕ), f n = ∑ j, ∑' (m : ℕ), f (j.val + N * m)
• AlgebraicGeometry.instT0SpaceCarrierCarrierCommRingCat 📋 Mathlib.AlgebraicGeometry.Properties
  (X : AlgebraicGeometry.Scheme) : T0Space ↥X
• CompletableTopField.toT0Space 📋 Mathlib.Topology.Algebra.UniformField
  {K : Type u_1} {inst✝ : Field K} {inst✝¹ : UniformSpace K} [self : CompletableTopField K] : T0Space K
• UniformSpace.Completion.instNontrivialOfT0Space 📋 Mathlib.Topology.Algebra.UniformField
  (K : Type u_1) [Field K] [UniformSpace K] [T0Space K] : Nontrivial (UniformSpace.Completion K)
• completableTopField_of_complete 📋 Mathlib.Topology.Algebra.UniformField
  (L : Type u_3) [Field L] [UniformSpace L] [IsTopologicalDivisionRing L] [T0Space L] [CompleteSpace L] : CompletableTopField L
• CompletableTopField.mk 📋 Mathlib.Topology.Algebra.UniformField
  {K : Type u_1} [Field K] [UniformSpace K] [toT0Space : T0Space K] (nice : ∀ (F : Filter K), Cauchy F → nhds 0 ⊓ F = ⊥ → Cauchy (Filter.map (fun x => x⁻¹) F)) : CompletableTopField K
• IsUniformInducing.completableTopField 📋 Mathlib.Topology.Algebra.UniformField
  {α : Type u_3} {β : Type u_4} [Field β] [b : UniformSpace β] [CompletableTopField β] [Field α] [UniformSpace α] [T0Space α] {f : α →+* β} (hf : IsUniformInducing ⇑f) : CompletableTopField α
• ValuedRing.separated 📋 Mathlib.Topology.Algebra.Valued.ValuedField
  {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [Valued K Γ₀] : T0Space K
• IsDedekindDomain.HeightOneSpectrum.adicCompletion.instT0Space 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
  {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) : T0Space (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)
• MvPowerSeries.WithPiTopology.instT0Space 📋 Mathlib.RingTheory.MvPowerSeries.PiTopology
  {σ : Type u_1} {R : Type u_2} [TopologicalSpace R] [T0Space R] : T0Space (MvPowerSeries σ R)
• PowerSeries.WithPiTopology.instT0Space 📋 Mathlib.RingTheory.PowerSeries.PiTopology
  (R : Type u_1) [TopologicalSpace R] [T0Space R] : T0Space (PowerSeries R)
• ContinuousMapZero.instT0Space 📋 Mathlib.Topology.ContinuousMap.ContinuousMapZero
  {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T0Space R] : T0Space (ContinuousMapZero X R)
• TopologicalSpace.vietoris.instT0SpaceSetOfT1Space 📋 Mathlib.Topology.Sets.VietorisTopology
  {α : Type u_1} [TopologicalSpace α] [T1Space α] : T0Space (Set α)
• TopologicalSpace.Compacts.instT0SpaceOfT1Space 📋 Mathlib.Topology.Sets.VietorisTopology
  {α : Type u_1} [TopologicalSpace α] [T1Space α] : T0Space (TopologicalSpace.Compacts α)
• TopologicalSpace.NonemptyCompacts.instT0SpaceOfT1Space 📋 Mathlib.Topology.Sets.VietorisTopology
  {α : Type u_1} [TopologicalSpace α] [T1Space α] : T0Space (TopologicalSpace.NonemptyCompacts α)
• TopologicalSpace.Closeds.instT0Space 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] : T0Space (TopologicalSpace.Closeds α)
• TopologicalSpace.Closeds.discreteUniformity_iff 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : DiscreteUniformity (TopologicalSpace.Closeds α) ↔ DiscreteUniformity α
• UniformSpace.hausdorff.isClosedEmbedding_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : Topology.IsClosedEmbedding fun x => {x}
• TopologicalSpace.Closeds.isUniformEmbedding_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : IsUniformEmbedding fun x => {x}
• TopologicalSpace.Closeds.uniformContinuous_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : UniformContinuous fun x => {x}
• TopologicalSpace.Closeds.continuous_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : Continuous fun x => {x}
• TopologicalSpace.Closeds.isClosedEmbedding_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : Topology.IsClosedEmbedding fun x => {x}
• TopologicalSpace.Closeds.isEmbedding_singleton 📋 Mathlib.Topology.UniformSpace.Closeds
  {α : Type u_1} [UniformSpace α] [T0Space α] : Topology.IsEmbedding fun x => {x}
• 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
• FiberBundle.t0Space 📋 Mathlib.Topology.FiberBundle.Basic
  {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] [T0Space F] (b : B) : T0Space (E b)
• AddCircle.isAddQuotientCoveringMap_zsmul 📋 Mathlib.Topology.Covering.AddCircle
  {𝕜 : Type u_1} [TopologicalSpace 𝕜] [Ring 𝕜] [IsTopologicalRing 𝕜] (p : 𝕜) [T0Space (AddCircle p)] {n : ℤ} (hn : IsUnit ↑n) : IsAddQuotientCoveringMap (fun x => n • x) ↥(zsmulAddGroupHom n).ker
• AddCircle.isAddQuotientCoveringMap_zsmul_of_ne_zero 📋 Mathlib.Topology.Covering.AddCircle
  {𝕜 : Type u_1} [TopologicalSpace 𝕜] [Ring 𝕜] [IsTopologicalRing 𝕜] (p : 𝕜) [T0Space (AddCircle p)] [Algebra ℚ 𝕜] (n : ℤ) [NeZero n] : IsAddQuotientCoveringMap (fun x => n • x) ↥(zsmulAddGroupHom n).ker
• AddCircle.isAddQuotientCoveringMap_nsmul_of_ne_zero 📋 Mathlib.Topology.Covering.AddCircle
  {𝕜 : Type u_1} [TopologicalSpace 𝕜] [Ring 𝕜] [IsTopologicalRing 𝕜] (p : 𝕜) [T0Space (AddCircle p)] [Algebra ℚ 𝕜] (n : ℕ) [NeZero n] : IsAddQuotientCoveringMap (fun x => n • x) ↥(nsmulAddMonoidHom n).ker
• AddCircle.isAddQuotientCoveringMap_nsmul 📋 Mathlib.Topology.Covering.AddCircle
  {𝕜 : Type u_1} [TopologicalSpace 𝕜] [Ring 𝕜] [IsTopologicalRing 𝕜] (p : 𝕜) [T0Space (AddCircle p)] {n : ℕ} (hn : IsUnit ↑n) : IsAddQuotientCoveringMap (fun x => n • x) ↥(nsmulAddMonoidHom n).ker
• Matrix.SpecialLinearGroup.isClosedEmbedding_toGL 📋 Mathlib.Topology.Algebra.Group.Matrix
  {n : Type u_1} {R : Type u_2} [Fintype n] [DecidableEq n] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [T0Space R] : Topology.IsClosedEmbedding ⇑Matrix.SpecialLinearGroup.toGL
• Topology.IsClosedEmbedding.generalLinearGroup_map 📋 Mathlib.Topology.Algebra.Group.Matrix
  {n : Type u_1} {R : Type u_2} {S : Type u_3} [Fintype n] [DecidableEq n] [CommRing R] [TopologicalSpace R] [CommRing S] [TopologicalSpace S] {f : R →+* S} [IsTopologicalRing R] [T0Space R] (hf : Topology.IsClosedEmbedding ⇑f) : Topology.IsClosedEmbedding ⇑(Matrix.GeneralLinearGroup.map f)
• OnePoint.instT0Space 📋 Mathlib.Topology.Compactification.OnePoint.Basic
  {X : Type u_1} [TopologicalSpace X] [T0Space X] : T0Space (OnePoint X)
• IsCompactOperator.restrict' 📋 Mathlib.Analysis.Normed.Operator.Compact.Basic
  {R₂ : Type u_2} [Semiring R₂] {M₂ : Type u_4} [UniformSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] [T0Space M₂] {f : M₂ →ₗ[R₂] M₂} (hf : IsCompactOperator ⇑f) {V : Submodule R₂ M₂} (hV : ∀ v ∈ V, f v ∈ V) [hcomplete : CompleteSpace ↥V] : IsCompactOperator ⇑(f.restrict hV)
• ContinuousLinearMap.isClosed_eigenspace 📋 Mathlib.LinearAlgebra.Eigenspace.ContinuousLinearMap
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [T0Space M] [ContinuousConstSMul R M] [IsTopologicalAddGroup M] (f : M →L[R] M) (μ : R) : IsClosed ↑(Module.End.eigenspace (↑f) μ)

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