ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.IsTopologicalBasis.secondCountableTopology
protected theorem IsTopologicalBasis.secondCountableTopology {b : Set (Set α)}
    (hb : IsTopologicalBasis b) (hc : b.Countable) : SecondCountableTopology α 
Arguments: {α : Type u} [t : TopologicalSpace α] {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) (hc : b.Countable)
Goal: SecondCountableTopology α
-- TopologicalSpace.SecondCountableTopology.mk'
lemma SecondCountableTopology.mk' {b : Set (Set α)} (hc : b.Countable) :
    @SecondCountableTopology α (generateFrom b) 
Arguments: {α : Type u} {b : Set (Set α)} (hc : b.Countable)
Goal: SecondCountableTopology α
-- TopologicalSpace.exists_countable_basis
theorem exists_countable_basis [SecondCountableTopology α] :
    ∃ b : Set (Set α), b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b 
Arguments: (α : Type u) [t : TopologicalSpace α] [SecondCountableTopology α]
Goal: ∃ b, b.Countable ∧ ∅ ∉ b ∧ TopologicalSpace.IsTopologicalBasis b
-- TopologicalSpace.countable_countableBasis
theorem countable_countableBasis [SecondCountableTopology α] : (countableBasis α).Countable 
Arguments: (α : Type u) [t : TopologicalSpace α] [SecondCountableTopology α]
Goal: (TopologicalSpace.countableBasis α).Countable
-- TopologicalSpace.empty_nmem_countableBasis
theorem empty_nmem_countableBasis [SecondCountableTopology α] : ∅ ∉ countableBasis α 
Arguments: (α : Type u) [t : TopologicalSpace α] [SecondCountableTopology α]
Goal: ∅ ∉ TopologicalSpace.countableBasis α
-- TopologicalSpace.isBasis_countableBasis
theorem isBasis_countableBasis [SecondCountableTopology α] :
    IsTopologicalBasis (countableBasis α) 
Arguments: (α : Type u) [t : TopologicalSpace α] [SecondCountableTopology α]
Goal: TopologicalSpace.IsTopologicalBasis (TopologicalSpace.countableBasis α)
-- TopologicalSpace.eq_generateFrom_countableBasis
theorem eq_generateFrom_countableBasis [SecondCountableTopology α] :
    ‹TopologicalSpace α› = generateFrom (countableBasis α) 
Arguments: (α : Type u) [t : TopologicalSpace α] [SecondCountableTopology α]
Goal: t = TopologicalSpace.generateFrom (TopologicalSpace.countableBasis α)
-- TopologicalSpace.isOpen_of_mem_countableBasis
theorem isOpen_of_mem_countableBasis [SecondCountableTopology α] {s : Set α}
    (hs : s ∈ countableBasis α) : IsOpen s 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {s : Set α} (hs : s ∈ TopologicalSpace.countableBasis α)
Goal: IsOpen s
-- TopologicalSpace.nonempty_of_mem_countableBasis
theorem nonempty_of_mem_countableBasis [SecondCountableTopology α] {s : Set α}
    (hs : s ∈ countableBasis α) : s.Nonempty 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {s : Set α} (hs : s ∈ TopologicalSpace.countableBasis α)
Goal: s.Nonempty
-- TopologicalSpace.secondCountableTopology_induced
theorem secondCountableTopology_induced (β) [t : TopologicalSpace β] [SecondCountableTopology β]
    (f : α → β) : @SecondCountableTopology α (t.induced f) 
Arguments: (α : Type u) (β : Type u_1) [t : TopologicalSpace β] [SecondCountableTopology β] (f : α → β)
Goal: SecondCountableTopology α
-- TopologicalSpace.secondCountableTopology_iInf
lemma secondCountableTopology_iInf {ι} [Countable ι] {t : ι → TopologicalSpace α}
    (ht : ∀ i, @SecondCountableTopology α (t i)) : @SecondCountableTopology α (⨅ i, t i) 
Arguments: {α : Type u} {ι : Sort u_1} [Countable ι] {t : ι → TopologicalSpace α} (ht : ∀ (i : ι), SecondCountableTopology α)
Goal: SecondCountableTopology α
-- TopologicalSpace.secondCountableTopology_of_countable_cover
theorem secondCountableTopology_of_countable_cover {ι} [Countable ι] {U : ι → Set α}
    [∀ i, SecondCountableTopology (U i)] (Uo : ∀ i, IsOpen (U i)) (hc : ⋃ i, U i = univ) :
    SecondCountableTopology α 
Arguments: {α : Type u} [t : TopologicalSpace α] {ι : Sort u_1} [Countable ι] {U : ι → Set α} [∀ (i : ι), SecondCountableTopology ↑(U i)] (Uo : ∀ (i : ι), IsOpen (U i)) (hc : ⋃ i, U i = Set.univ)
Goal: SecondCountableTopology α
-- TopologicalSpace.isOpen_iUnion_countable
theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
    (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {ι : Type u_1} (s : ι → Set α) (H : ∀ (i : ι), IsOpen (s i))
Goal: ∃ T, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i
-- TopologicalSpace.isOpen_biUnion_countable
theorem isOpen_biUnion_countable [SecondCountableTopology α] {ι : Type*} (I : Set ι) (s : ι → Set α)
    (H : ∀ i ∈ I, IsOpen (s i)) : ∃ T ⊆ I, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i ∈ I, s i 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {ι : Type u_1} (I : Set ι) (s : ι → Set α) (H : ∀ i ∈ I, IsOpen (s i))
Goal: ∃ T ⊆ I, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i ∈ I, s i
-- TopologicalSpace.isOpen_sUnion_countable
theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
    (H : ∀ s ∈ S, IsOpen s) : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] (S : Set (Set α)) (H : ∀ s ∈ S, IsOpen s)
Goal: ∃ T, T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S
-- TopologicalSpace.countable_cover_nhds
theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (hf : ∀ x, f x ∈ 𝓝 x) :
    ∃ s : Set α, s.Countable ∧ ⋃ x ∈ s, f x = univ 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {f : α → Set α} (hf : ∀ (x : α), f x ∈ 𝓝 x)
Goal: ∃ s, s.Countable ∧ ⋃ x ∈ s, f x = Set.univ
-- TopologicalSpace.countable_cover_nhdsWithin
theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
    (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t ⊆ s, t.Countable ∧ s ⊆ ⋃ x ∈ t, f x 
Arguments: {α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] {f : α → Set α} {s : Set α} (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x)
Goal: ∃ t ⊆ s, t.Countable ∧ s ⊆ ⋃ x ∈ t, f x
-- TopologicalSpace.IsTopologicalBasis.sigma
theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
    (hs : ∀ i, IsTopologicalBasis (s i)) :
    IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → TopologicalSpace (E i)] {s : (i : ι) → Set (Set (E i))} (hs : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (s i))
Goal: TopologicalSpace.IsTopologicalBasis (⋃ i, (fun u => Sigma.mk i '' u) '' s i)
-- TopologicalSpace.IsTopologicalBasis.sum
theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t : Set (Set β)}
    (ht : IsTopologicalBasis t) :
    IsTopologicalBasis ((fun u => Sum.inl '' u) '' s ∪ (fun u => Sum.inr '' u) '' t) 
Arguments: {α : Type u} {β : Type u_1} [TopologicalSpace α] [TopologicalSpace β] {s : Set (Set α)} (hs : TopologicalSpace.IsTopologicalBasis s) {t : Set (Set β)} (ht : TopologicalSpace.IsTopologicalBasis t)
Goal: TopologicalSpace.IsTopologicalBasis ((fun u => Sum.inl '' u) '' s ∪ (fun u => Sum.inr '' u) '' t)
-- TopologicalSpace.IsTopologicalBasis.quotientMap
theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
    (h' : QuotientMap π) (h : IsOpenMap π) : IsTopologicalBasis (Set.image π '' V) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] {π : X → Y} {V : Set (Set X)} (hV : TopologicalSpace.IsTopologicalBasis V) (h' : QuotientMap π) (h : IsOpenMap π)
Goal: TopologicalSpace.IsTopologicalBasis (Set.image π '' V)
-- QuotientMap.secondCountableTopology
theorem _root_.QuotientMap.secondCountableTopology [SecondCountableTopology X] (h' : QuotientMap π)
    (h : IsOpenMap π) : SecondCountableTopology Y 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] {π : X → Y} [SecondCountableTopology X] (h' : QuotientMap π) (h : IsOpenMap π)
Goal: SecondCountableTopology Y
-- TopologicalSpace.IsTopologicalBasis.quotient
theorem IsTopologicalBasis.quotient {V : Set (Set X)} (hV : IsTopologicalBasis V)
    (h : IsOpenMap (Quotient.mk' : X → Quotient S)) :
    IsTopologicalBasis (Set.image (Quotient.mk' : X → Quotient S) '' V) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Setoid X} {V : Set (Set X)} (hV : TopologicalSpace.IsTopologicalBasis V) (h : IsOpenMap Quotient.mk')
Goal: TopologicalSpace.IsTopologicalBasis (Set.image Quotient.mk' '' V)
-- TopologicalSpace.Quotient.secondCountableTopology
theorem Quotient.secondCountableTopology [SecondCountableTopology X]
    (h : IsOpenMap (Quotient.mk' : X → Quotient S)) : SecondCountableTopology (Quotient S) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S : Setoid X} [SecondCountableTopology X] (h : IsOpenMap Quotient.mk')
Goal: SecondCountableTopology (Quotient S)
-- Inducing.secondCountableTopology
protected theorem Inducing.secondCountableTopology [SecondCountableTopology β] (hf : Inducing f) :
    SecondCountableTopology α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} [SecondCountableTopology β] (hf : Inducing f)
Goal: SecondCountableTopology α
-- Embedding.secondCountableTopology
protected theorem Embedding.secondCountableTopology [SecondCountableTopology β] (hf : Embedding f) :
    SecondCountableTopology α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} [SecondCountableTopology β] (hf : Embedding f)
Goal: SecondCountableTopology α
-- Embedding.separableSpace
protected theorem Embedding.separableSpace [TopologicalSpace α]
    [TopologicalSpace β] [SecondCountableTopology β] {f : α → β} (hf : Embedding f) :
    TopologicalSpace.SeparableSpace α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] {f : α → β} (hf : Embedding f)
Goal: TopologicalSpace.SeparableSpace α
-- TopCat.id_app
theorem id_app (X : TopCat.{u}) (x : ↑X) : (𝟙 X : X ⟶ X) x = x 
Arguments: (X : TopCat) (x : ↑X)
Goal: (𝟙 X) x = x
-- TopCat.comp_app
theorem comp_app {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
    (f ≫ g : X → Z) x = g (f x) 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↑X)
Goal: (f ≫ g) x = g (f x)
-- TopCat.coe_id
@[simp] theorem coe_id (X : TopCat.{u}) : (𝟙 X : X → X) = id 
Arguments: (X : TopCat)
Goal: ⇑(𝟙 X) = id
-- TopCat.coe_comp
@[simp] theorem coe_comp {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f ≫ g : X → Z) = g ∘ f 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: ⇑(f ≫ g) = ⇑g ∘ ⇑f
-- TopCat.hom_inv_id_apply
@[simp]
lemma hom_inv_id_apply {X Y : TopCat} (f : X ≅ Y) (x : X) : f.inv (f.hom x) = x 
Arguments: {X Y : TopCat} (f : X ≅ Y) (x : ↑X)
Goal: f.inv (f.hom x) = x
-- TopCat.inv_hom_id_apply
@[simp]
lemma inv_hom_id_apply {X Y : TopCat} (f : X ≅ Y) (y : Y) : f.hom (f.inv y) = y 
Arguments: {X Y : TopCat} (f : X ≅ Y) (y : ↑Y)
Goal: f.hom (f.inv y) = y
-- TopCat.coe_of
@[simp]
theorem coe_of (X : Type u) [TopologicalSpace X] : (of X : Type u) = X 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: ↑(TopCat.of X) = X
-- TopCat.coe_of_of
@[simp] theorem coe_of_of {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y]
    {f : C(X, Y)} {x} :
    @DFunLike.coe (TopCat.of X ⟶ TopCat.of Y) ((CategoryTheory.forget TopCat).obj (TopCat.of X))
      (fun _ ↦ (CategoryTheory.forget TopCat).obj (TopCat.of Y)) ConcreteCategory.instFunLike
      f x =
    @DFunLike.coe C(X, Y) X
      (fun _ ↦ Y) _
      f x 
Arguments: {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {x : (CategoryTheory.forget TopCat).obj (TopCat.of X)}
Goal: f x = f x
-- TopCat.hom_apply
lemma hom_apply {X Y : TopCat} (f : X ⟶ Y) (x : X) : f x = ContinuousMap.toFun f x 
Arguments: {X Y : TopCat} (f : X ⟶ Y) (x : ↑X)
Goal: f x = f.toFun x
-- TopCat.isoOfHomeo_hom
@[simps]
def isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : X ≅ Y 
Arguments: {X Y : TopCat} (f : ↑X ≃ₜ ↑Y)
Goal: (TopCat.isoOfHomeo f).hom = f.toContinuousMap
-- TopCat.isoOfHomeo_inv
@[simps]
def isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : X ≅ Y 
Arguments: {X Y : TopCat} (f : ↑X ≃ₜ ↑Y)
Goal: (TopCat.isoOfHomeo f).inv = f.symm.toContinuousMap
-- TopCat.homeoOfIso_apply
@[simps]
def homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : X ≃ₜ Y 
Arguments: {X Y : TopCat} (f : X ≅ Y) (a : ↑X)
Goal: (TopCat.homeoOfIso f) a = f.hom a
-- TopCat.homeoOfIso_symm_apply
@[simps]
def homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : X ≃ₜ Y 
Arguments: {X Y : TopCat} (f : X ≅ Y) (a : ↑Y)
Goal: (TopCat.homeoOfIso f).symm a = f.inv a
-- TopCat.of_isoOfHomeo
@[simp]
theorem of_isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : homeoOfIso (isoOfHomeo f) = f 
Arguments: {X Y : TopCat} (f : ↑X ≃ₜ ↑Y)
Goal: TopCat.homeoOfIso (TopCat.isoOfHomeo f) = f
-- TopCat.of_homeoOfIso
@[simp]
theorem of_homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : isoOfHomeo (homeoOfIso f) = f 
Arguments: {X Y : TopCat} (f : X ≅ Y)
Goal: TopCat.isoOfHomeo (TopCat.homeoOfIso f) = f
-- TopCat.openEmbedding_iff_comp_isIso
theorem openEmbedding_iff_comp_isIso {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] :
    OpenEmbedding (f ≫ g) ↔ OpenEmbedding f 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso g]
Goal: OpenEmbedding ⇑(f ≫ g) ↔ OpenEmbedding ⇑f
-- TopCat.openEmbedding_iff_comp_isIso'
@[simp]
theorem openEmbedding_iff_comp_isIso' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] :
    OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding f 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso g]
Goal: OpenEmbedding ((CategoryTheory.forget TopCat).map f ≫ (CategoryTheory.forget TopCat).map g) ↔ OpenEmbedding ⇑f
-- TopCat.openEmbedding_iff_isIso_comp
theorem openEmbedding_iff_isIso_comp {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] :
    OpenEmbedding (f ≫ g) ↔ OpenEmbedding g 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso f]
Goal: OpenEmbedding ⇑(f ≫ g) ↔ OpenEmbedding ⇑g
-- TopCat.openEmbedding_iff_isIso_comp'
@[simp]
theorem openEmbedding_iff_isIso_comp' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] :
    OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding g 
Arguments: {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso f]
Goal: OpenEmbedding ((CategoryTheory.forget TopCat).map f ≫ (CategoryTheory.forget TopCat).map g) ↔ OpenEmbedding ⇑g
-- IsLindelof.compl_mem_sets
theorem IsLindelof.compl_mem_sets (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f]
    (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) : sᶜ ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f] (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f)
Goal: sᶜ ∈ f
-- IsLindelof.compl_mem_sets_of_nhdsWithin
theorem IsLindelof.compl_mem_sets_of_nhdsWithin (hs : IsLindelof s) {f : Filter X}
    [CountableInterFilter f] (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f] (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f)
Goal: sᶜ ∈ f
-- IsLindelof.induction_on
@[elab_as_elim]
theorem IsLindelof.induction_on (hs : IsLindelof s) {p : Set X → Prop}
    (hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s)
    (hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S))
    (hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {p : Set X → Prop} (hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s) (hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S)) (hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t)
Goal: p s
-- IsLindelof.inter_right
theorem IsLindelof.inter_right (hs : IsLindelof s) (ht : IsClosed t) : IsLindelof (s ∩ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsLindelof s) (ht : IsClosed t)
Goal: IsLindelof (s ∩ t)
-- IsLindelof.inter_left
theorem IsLindelof.inter_left (ht : IsLindelof t) (hs : IsClosed s) : IsLindelof (s ∩ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (ht : IsLindelof t) (hs : IsClosed s)
Goal: IsLindelof (s ∩ t)
-- IsLindelof.diff
theorem IsLindelof.diff (hs : IsLindelof s) (ht : IsOpen t) : IsLindelof (s \ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsLindelof s) (ht : IsOpen t)
Goal: IsLindelof (s \ t)
-- IsLindelof.of_isClosed_subset
theorem IsLindelof.of_isClosed_subset (hs : IsLindelof s) (ht : IsClosed t) (h : t ⊆ s) :
    IsLindelof t 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsLindelof s) (ht : IsClosed t) (h : t ⊆ s)
Goal: IsLindelof t
-- IsLindelof.image_of_continuousOn
theorem IsLindelof.image_of_continuousOn {f : X → Y} (hs : IsLindelof s) (hf : ContinuousOn f s) :
    IsLindelof (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hs : IsLindelof s) (hf : ContinuousOn f s)
Goal: IsLindelof (f '' s)
-- IsLindelof.image
theorem IsLindelof.image {f : X → Y} (hs : IsLindelof s) (hf : Continuous f) :
    IsLindelof (f '' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hs : IsLindelof s) (hf : Continuous f)
Goal: IsLindelof (f '' s)
-- IsLindelof.adherence_nhdset
theorem IsLindelof.adherence_nhdset {f : Filter X} [CountableInterFilter f] (hs : IsLindelof s)
    (hf₂ : f ≤ 𝓟 s) (ht₁ : IsOpen t) (ht₂ : ∀ x ∈ s, ClusterPt x f → x ∈ t) : t ∈ f 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} {f : Filter X} [CountableInterFilter f] (hs : IsLindelof s) (hf₂ : f ≤ 𝓟 s) (ht₁ : IsOpen t) (ht₂ : ∀ x ∈ s, ClusterPt x f → x ∈ t)
Goal: t ∈ f
-- IsLindelof.elim_countable_subcover
theorem IsLindelof.elim_countable_subcover {ι : Type v} (hs : IsLindelof s) (U : ι → Set X)
    (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) :
    ∃ r : Set ι, r.Countable ∧ (s ⊆ ⋃ i ∈ r, U i) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (U : ι → Set X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i)
Goal: ∃ r, r.Countable ∧ s ⊆ ⋃ i ∈ r, U i
-- IsLindelof.elim_nhds_subcover'
theorem IsLindelof.elim_nhds_subcover' (hs : IsLindelof s) (U : ∀ x ∈ s, Set X)
    (hU : ∀ x (hx : x ∈ s), U x ‹x ∈ s› ∈ 𝓝 x) :
    ∃ t : Set s, t.Countable ∧ s ⊆ ⋃ x ∈ t, U (x : s) x.2 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (U : (x : X) → x ∈ s → Set X) (hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x)
Goal: ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, U ↑x ⋯
-- IsLindelof.elim_nhds_subcover
theorem IsLindelof.elim_nhds_subcover (hs : IsLindelof s) (U : X → Set X)
    (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
    ∃ t : Set X, t.Countable ∧ (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (U : X → Set X) (hU : ∀ x ∈ s, U x ∈ 𝓝 x)
Goal: ∃ t, t.Countable ∧ (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x
-- IsLindelof.indexed_countable_subcover
theorem IsLindelof.indexed_countable_subcover {ι : Type v} [Nonempty ι]
    (hs : IsLindelof s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) :
    ∃ f : ℕ → ι, s ⊆ ⋃ n, U (f n) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} [Nonempty ι] (hs : IsLindelof s) (U : ι → Set X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i)
Goal: ∃ f, s ⊆ ⋃ n, U (f n)
-- IsLindelof.disjoint_nhdsSet_left
theorem IsLindelof.disjoint_nhdsSet_left {l : Filter X} [CountableInterFilter l]
    (hs : IsLindelof s) :
    Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} [CountableInterFilter l] (hs : IsLindelof s)
Goal: Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l
-- IsLindelof.disjoint_nhdsSet_right
theorem IsLindelof.disjoint_nhdsSet_right {l : Filter X} [CountableInterFilter l]
    (hs : IsLindelof s) : Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} [CountableInterFilter l] (hs : IsLindelof s)
Goal: Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)
-- IsLindelof.elim_countable_subfamily_closed
theorem IsLindelof.elim_countable_subfamily_closed {ι : Type v} (hs : IsLindelof s)
    (t : ι → Set X) (htc : ∀ i, IsClosed (t i)) (hst : (s ∩ ⋂ i, t i) = ∅) :
    ∃ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (t : ι → Set X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : s ∩ ⋂ i, t i = ∅)
Goal: ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
-- IsLindelof.inter_iInter_nonempty
theorem IsLindelof.inter_iInter_nonempty {ι : Type v} (hs : IsLindelof s) (t : ι → Set X)
    (htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i).Nonempty) :
    (s ∩ ⋂ i, t i).Nonempty 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (t : ι → Set X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : ∀ (u : Set ι), u.Countable ∧ (s ∩ ⋂ i ∈ u, t i).Nonempty)
Goal: (s ∩ ⋂ i, t i).Nonempty
-- IsLindelof.elim_countable_subcover_image
theorem IsLindelof.elim_countable_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsLindelof s)
    (hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
    ∃ b', b' ⊆ b ∧ Set.Countable b' ∧ s ⊆ ⋃ i ∈ b', c i 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set X} {b : Set ι} {c : ι → Set X} (hs : IsLindelof s) (hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i)
Goal: ∃ b' ⊆ b, b'.Countable ∧ s ⊆ ⋃ i ∈ b', c i
-- isLindelof_of_countable_subcover
theorem isLindelof_of_countable_subcover
    (h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
    ∃ t : Set ι, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (h : ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i)
Goal: IsLindelof s
-- isLindelof_of_countable_subfamily_closed
theorem isLindelof_of_countable_subfamily_closed
    (h :
      ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ →
        ∃ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i) = ∅) :
    IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅)
Goal: IsLindelof s
-- isLindelof_iff_countable_subcover
theorem isLindelof_iff_countable_subcover :
    IsLindelof s ↔ ∀ {ι : Type u} (U : ι → Set X),
      (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) → ∃ t : Set ι, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsLindelof s ↔
  ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i
-- isLindelof_iff_countable_subfamily_closed
theorem isLindelof_iff_countable_subfamily_closed :
    IsLindelof s ↔ ∀ {ι : Type u} (t : ι → Set X),
    (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅
    → ∃ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsLindelof s ↔
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
-- isLindelof_empty
@[simp]
theorem isLindelof_empty : IsLindelof (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsLindelof ∅
-- isLindelof_singleton
@[simp]
theorem isLindelof_singleton {x : X} : IsLindelof ({x} : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {x : X}
Goal: IsLindelof {x}
-- Set.Subsingleton.isLindelof
theorem Set.Subsingleton.isLindelof (hs : s.Subsingleton) : IsLindelof s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Subsingleton)
Goal: IsLindelof s
-- Set.Countable.isLindelof_biUnion
theorem Set.Countable.isLindelof_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Countable)
    (hf : ∀ i ∈ s, IsLindelof (f i)) : IsLindelof (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set ι} {f : ι → Set X} (hs : s.Countable) (hf : ∀ i ∈ s, IsLindelof (f i))
Goal: IsLindelof (⋃ i ∈ s, f i)
-- Set.Finite.isLindelof_biUnion
theorem Set.Finite.isLindelof_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Finite)
    (hf : ∀ i ∈ s, IsLindelof (f i)) : IsLindelof (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set ι} {f : ι → Set X} (hs : s.Finite) (hf : ∀ i ∈ s, IsLindelof (f i))
Goal: IsLindelof (⋃ i ∈ s, f i)