ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
    (ou : IsOpen u) : ∃ S ⊆ B, u = ⋃₀ S 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (ou : IsOpen u)
Goal: ∃ S ⊆ B, u = ⋃₀ S
-- TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
    {u : Set α} : IsOpen u ↔ ∃ S ⊆ B, u = ⋃₀ S 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α}
Goal: IsOpen u ↔ ∃ S ⊆ B, u = ⋃₀ S
-- TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
    (ou : IsOpen u) : ∃ (β : Type u) (f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (ou : IsOpen u)
Goal: ∃ β f, u = ⋃ i, f i ∧ ∀ (i : β), f i ∈ B
-- TopologicalSpace.IsTopologicalBasis.subset_of_forall_subset
lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologicalBasis B)
    (hs : IsOpen s) (h : ∀ U ∈ B, U ⊆ s → U ⊆ t) : s ⊆ t 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} {s : Set α}
Goal: ∀ {t_1 : Set α}, TopologicalSpace.IsTopologicalBasis B → IsOpen s → (∀ U ∈ B, U ⊆ s → U ⊆ t_1) → s ⊆ t_1
-- TopologicalSpace.IsTopologicalBasis.eq_of_forall_subset_iff
lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)
    (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} {s : Set α}
Goal: ∀ {t_1 : Set α}, TopologicalSpace.IsTopologicalBasis B → IsOpen s → IsOpen t_1 → (∀ U ∈ B, U ⊆ s ↔ U ⊆ t_1) → s = t_1
-- TopologicalSpace.IsTopologicalBasis.mem_closure_iff
theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
    {a : α} : a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty 
Arguments: {α : Type u} [t : TopologicalSpace α] {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) {s : Set α} {a : α}
Goal: a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty
-- TopologicalSpace.IsTopologicalBasis.dense_iff
theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α} :
    Dense s ↔ ∀ o ∈ b, Set.Nonempty o → (o ∩ s).Nonempty 
Arguments: {α : Type u} [t : TopologicalSpace α] {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) {s : Set α}
Goal: Dense s ↔ ∀ o ∈ b, o.Nonempty → (o ∩ s).Nonempty
-- TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Set α)}
    (hB : IsTopologicalBasis B) {f : α → β} : IsOpenMap f ↔ ∀ s ∈ B, IsOpen (f '' s) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {f : α → β}
Goal: IsOpenMap f ↔ ∀ s ∈ B, IsOpen (f '' s)
-- TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTopologicalBasis B)
    {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (hu : u.Nonempty) (ou : IsOpen u)
Goal: ∃ v ∈ B, v.Nonempty ∧ v ⊆ u
-- TopologicalSpace.isTopologicalBasis_opens
theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U } 
Arguments: {α : Type u} [t : TopologicalSpace α]
Goal: TopologicalSpace.IsTopologicalBasis {U | IsOpen U}
-- TopologicalSpace.IsTopologicalBasis.inducing
protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
    (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis ((preimage f) '' T) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {f : α → β} {T : Set (Set β)} (hf : Inducing f) (h : TopologicalSpace.IsTopologicalBasis T)
Goal: TopologicalSpace.IsTopologicalBasis (Set.preimage f '' T)
-- TopologicalSpace.IsTopologicalBasis.induced
protected theorem IsTopologicalBasis.induced [s : TopologicalSpace β] (f : α → β)
    {T : Set (Set β)} (h : IsTopologicalBasis T) :
    IsTopologicalBasis (t := induced f s) ((preimage f) '' T) 
Arguments: {α : Type u} {β : Type u_1} [s : TopologicalSpace β] (f : α → β) {T : Set (Set β)} (h : TopologicalSpace.IsTopologicalBasis T)
Goal: TopologicalSpace.IsTopologicalBasis (Set.preimage f '' T)
-- TopologicalSpace.IsTopologicalBasis.inf
protected theorem IsTopologicalBasis.inf {t₁ t₂ : TopologicalSpace β} {B₁ B₂ : Set (Set β)}
    (h₁ : IsTopologicalBasis (t := t₁) B₁) (h₂ : IsTopologicalBasis (t := t₂) B₂) :
    IsTopologicalBasis (t := t₁ ⊓ t₂) (image2 (· ∩ ·) B₁ B₂) 
Arguments: {β : Type u_1} {t₁ t₂ : TopologicalSpace β} {B₁ B₂ : Set (Set β)} (h₁ : TopologicalSpace.IsTopologicalBasis B₁) (h₂ : TopologicalSpace.IsTopologicalBasis B₂)
Goal: TopologicalSpace.IsTopologicalBasis (Set.image2 (fun x x_1 => x ∩ x_1) B₁ B₂)
-- TopologicalSpace.IsTopologicalBasis.inf_induced
theorem IsTopologicalBasis.inf_induced {γ} [s : TopologicalSpace β] {B₁ : Set (Set α)}
    {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) (f₁ : γ → α)
    (f₂ : γ → β) :
    IsTopologicalBasis (t := induced f₁ t ⊓ induced f₂ s) (image2 (f₁ ⁻¹' · ∩ f₂ ⁻¹' ·) B₁ B₂) 
Arguments: {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)
Goal: TopologicalSpace.IsTopologicalBasis (Set.image2 (fun x x_1 => f₁ ⁻¹' x ∩ f₂ ⁻¹' x_1) B₁ B₂)
-- TopologicalSpace.IsTopologicalBasis.prod
protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
    {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
    IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {B₁ : Set (Set α)} {B₂ : Set (Set β)} (h₁ : TopologicalSpace.IsTopologicalBasis B₁) (h₂ : TopologicalSpace.IsTopologicalBasis B₂)
Goal: TopologicalSpace.IsTopologicalBasis (Set.image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂)
-- TopologicalSpace.isTopologicalBasis_of_cover
theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
    (Uc : ⋃ i, U i = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
    IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) 
Arguments: {α : Type u} [t : TopologicalSpace α] {ι : Sort u_2} {U : ι → Set α} (Uo : ∀ (i : ι), IsOpen (U i)) (Uc : ⋃ i, U i = Set.univ) {b : (i : ι) → Set (Set ↑(U i))} (hb : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (b i))
Goal: TopologicalSpace.IsTopologicalBasis (⋃ i, Set.image Subtype.val '' b i)
-- TopologicalSpace.IsTopologicalBasis.continuous_iff
protected theorem IsTopologicalBasis.continuous_iff {β : Type*} [TopologicalSpace β]
    {B : Set (Set β)} (hB : IsTopologicalBasis B) {f : α → β} :
    Continuous f ↔ ∀ s ∈ B, IsOpen (f ⁻¹' s) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {B : Set (Set β)} (hB : TopologicalSpace.IsTopologicalBasis B) {f : α → β}
Goal: Continuous f ↔ ∀ s ∈ B, IsOpen (f ⁻¹' s)
-- TopologicalSpace.IsTopologicalBasis.continuous
@[deprecated (since := "2023-12-24")]
protected theorem IsTopologicalBasis.continuous {β : Type*} [TopologicalSpace β] {B : Set (Set β)}
    (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {B : Set (Set β)} (hB : TopologicalSpace.IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s))
Goal: Continuous f
-- TopologicalSpace.separableSpace_iff
@[mk_iff] class SeparableSpace : Prop where
    exists_countable_dense : ∃ s : Set α, s.Countable ∧ Dense s
Arguments: (α : Type u) [t : TopologicalSpace α]
Goal: TopologicalSpace.SeparableSpace α ↔ ∃ s, s.Countable ∧ Dense s
-- TopologicalSpace.SeparableSpace.exists_countable_dense
@[mk_iff] class SeparableSpace : Prop where
    exists_countable_dense : ∃ s : Set α, s.Countable ∧ Dense s
Arguments: {α : Type u} [t : TopologicalSpace α] [self : TopologicalSpace.SeparableSpace α]
Goal: ∃ s, s.Countable ∧ Dense s
-- TopologicalSpace.exists_countable_dense
theorem exists_countable_dense [SeparableSpace α] : ∃ s : Set α, s.Countable ∧ Dense s 
Arguments: (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α]
Goal: ∃ s, s.Countable ∧ Dense s
-- TopologicalSpace.exists_dense_seq
theorem exists_dense_seq [SeparableSpace α] [Nonempty α] : ∃ u : ℕ → α, DenseRange u 
Arguments: (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [Nonempty α]
Goal: ∃ u, DenseRange u
-- TopologicalSpace.denseRange_denseSeq
@[simp]
theorem denseRange_denseSeq [SeparableSpace α] [Nonempty α] : DenseRange (denseSeq α) 
Arguments: (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [Nonempty α]
Goal: DenseRange (TopologicalSpace.denseSeq α)
-- TopologicalSpace.SeparableSpace.of_denseRange
theorem SeparableSpace.of_denseRange {ι : Sort _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
    SeparableSpace α 
Arguments: {α : Type u} [t : TopologicalSpace α] {ι : Type u_2} [Countable ι] (u : ι → α) (hu : DenseRange u)
Goal: TopologicalSpace.SeparableSpace α
-- DenseRange.separableSpace
protected theorem _root_.DenseRange.separableSpace [SeparableSpace α] [TopologicalSpace β]
    {f : α → β} (h : DenseRange f) (h' : Continuous f) : SeparableSpace β 
Arguments: {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [TopologicalSpace β] {f : α → β} (h : DenseRange f) (h' : Continuous f)
Goal: TopologicalSpace.SeparableSpace β
-- QuotientMap.separableSpace
theorem _root_.QuotientMap.separableSpace [SeparableSpace α] [TopologicalSpace β] {f : α → β}
    (hf : QuotientMap f) : SeparableSpace β 
Arguments: {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [TopologicalSpace β] {f : α → β} (hf : QuotientMap f)
Goal: TopologicalSpace.SeparableSpace β
-- TopologicalSpace.separableSpace_iff_countable
theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α ↔ Countable α 
Arguments: {α : Type u} [t : TopologicalSpace α] [DiscreteTopology α]
Goal: TopologicalSpace.SeparableSpace α ↔ Countable α
-- Pairwise.countable_of_isOpen_disjoint
theorem _root_.Pairwise.countable_of_isOpen_disjoint [SeparableSpace α] {ι : Type*}
    {s : ι → Set α} (hd : Pairwise (Disjoint on s)) (ho : ∀ i, IsOpen (s i))
    (hne : ∀ i, (s i).Nonempty) : Countable ι 
Arguments: {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} (hd : Pairwise (Disjoint on s)) (ho : ∀ (i : ι), IsOpen (s i)) (hne : ∀ (i : ι), (s i).Nonempty)
Goal: Countable ι
-- Set.PairwiseDisjoint.countable_of_isOpen
theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}
    {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ho : ∀ i ∈ a, IsOpen (s i))
    (hne : ∀ i ∈ a, (s i).Nonempty) : a.Countable 
Arguments: {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ho : ∀ i ∈ a, IsOpen (s i)) (hne : ∀ i ∈ a, (s i).Nonempty)
Goal: a.Countable
-- Set.PairwiseDisjoint.countable_of_nonempty_interior
theorem _root_.Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpace α] {ι : Type*}
    {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s)
    (ha : ∀ i ∈ a, (interior (s i)).Nonempty) : a.Countable 
Arguments: {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, (interior (s i)).Nonempty)
Goal: a.Countable
-- TopologicalSpace.IsSeparable.mono
theorem IsSeparable.mono {s u : Set α} (hs : IsSeparable s) (hu : u ⊆ s) : IsSeparable u 
Arguments: {α : Type u} [t : TopologicalSpace α] {s u : Set α} (hs : TopologicalSpace.IsSeparable s) (hu : u ⊆ s)
Goal: TopologicalSpace.IsSeparable u
-- TopologicalSpace.IsSeparable.iUnion
theorem IsSeparable.iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
    (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) 
Arguments: {α : Type u} [t : TopologicalSpace α] {ι : Sort u_2} [Countable ι] {s : ι → Set α} (hs : ∀ (i : ι), TopologicalSpace.IsSeparable (s i))
Goal: TopologicalSpace.IsSeparable (⋃ i, s i)
-- TopologicalSpace.isSeparable_iUnion
@[simp]
theorem isSeparable_iUnion {ι : Sort*} [Countable ι] {s : ι → Set α} :
    IsSeparable (⋃ i, s i) ↔ ∀ i, IsSeparable (s i) 
Arguments: {α : Type u} [t : TopologicalSpace α] {ι : Sort u_2} [Countable ι] {s : ι → Set α}
Goal: TopologicalSpace.IsSeparable (⋃ i, s i) ↔ ∀ (i : ι), TopologicalSpace.IsSeparable (s i)
-- TopologicalSpace.isSeparable_union
@[simp]
theorem isSeparable_union {s t : Set α} : IsSeparable (s ∪ t) ↔ IsSeparable s ∧ IsSeparable t 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α}
Goal: ∀ {t_1 : Set α},
  TopologicalSpace.IsSeparable (s ∪ t_1) ↔ TopologicalSpace.IsSeparable s ∧ TopologicalSpace.IsSeparable t_1
-- TopologicalSpace.IsSeparable.union
theorem IsSeparable.union {s u : Set α} (hs : IsSeparable s) (hu : IsSeparable u) :
    IsSeparable (s ∪ u) 
Arguments: {α : Type u} [t : TopologicalSpace α] {s u : Set α} (hs : TopologicalSpace.IsSeparable s) (hu : TopologicalSpace.IsSeparable u)
Goal: TopologicalSpace.IsSeparable (s ∪ u)
-- TopologicalSpace.isSeparable_closure
@[simp]
theorem isSeparable_closure : IsSeparable (closure s) ↔ IsSeparable s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α}
Goal: TopologicalSpace.IsSeparable (closure s) ↔ TopologicalSpace.IsSeparable s
-- Set.Countable.isSeparable
theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α} (hs : s.Countable)
Goal: TopologicalSpace.IsSeparable s
-- Set.Finite.isSeparable
theorem _root_.Set.Finite.isSeparable {s : Set α} (hs : s.Finite) : IsSeparable s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α} (hs : s.Finite)
Goal: TopologicalSpace.IsSeparable s
-- TopologicalSpace.IsSeparable.univ_pi
theorem IsSeparable.univ_pi {ι : Type*} [Countable ι] {X : ι → Type*} {s : ∀ i, Set (X i)}
    [∀ i, TopologicalSpace (X i)] (h : ∀ i, IsSeparable (s i)) :
    IsSeparable (univ.pi s) 
Arguments: {ι : Type u_2} [Countable ι] {X : ι → Type u_3} {s : (i : ι) → Set (X i)} [(i : ι) → TopologicalSpace (X i)] (h : ∀ (i : ι), TopologicalSpace.IsSeparable (s i))
Goal: TopologicalSpace.IsSeparable (Set.univ.pi s)
-- TopologicalSpace.isSeparable_pi
lemma isSeparable_pi {ι : Type*} [Countable ι] {α : ι → Type*} {s : ∀ i, Set (α i)}
    [∀ i, TopologicalSpace (α i)] (h : ∀ i, IsSeparable (s i)) :
    IsSeparable {f : ∀ i, α i | ∀ i, f i ∈ s i} 
Arguments: {ι : Type u_2} [Countable ι] {α : ι → Type u_3} {s : (i : ι) → Set (α i)} [(i : ι) → TopologicalSpace (α i)] (h : ∀ (i : ι), TopologicalSpace.IsSeparable (s i))
Goal: TopologicalSpace.IsSeparable {f | ∀ (i : ι), f i ∈ s i}
-- TopologicalSpace.IsSeparable.prod
lemma IsSeparable.prod {β : Type*} [TopologicalSpace β]
    {s : Set α} {t : Set β} (hs : IsSeparable s) (ht : IsSeparable t) :
    IsSeparable (s ×ˢ t) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {s : Set α}
Goal: ∀ {t_1 : Set β},
  TopologicalSpace.IsSeparable s → TopologicalSpace.IsSeparable t_1 → TopologicalSpace.IsSeparable (s ×ˢ t_1)
-- TopologicalSpace.IsSeparable.image
theorem IsSeparable.image {β : Type*} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
    {f : α → β} (hf : Continuous f) : IsSeparable (f '' s) 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] {s : Set α} (hs : TopologicalSpace.IsSeparable s) {f : α → β} (hf : Continuous f)
Goal: TopologicalSpace.IsSeparable (f '' s)
-- Dense.isSeparable_iff
theorem _root_.Dense.isSeparable_iff (hs : Dense s) :
    IsSeparable s ↔ SeparableSpace α 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α} (hs : Dense s)
Goal: TopologicalSpace.IsSeparable s ↔ TopologicalSpace.SeparableSpace α
-- TopologicalSpace.isSeparable_univ_iff
theorem isSeparable_univ_iff : IsSeparable (univ : Set α) ↔ SeparableSpace α 
Arguments: {α : Type u} [t : TopologicalSpace α]
Goal: TopologicalSpace.IsSeparable Set.univ ↔ TopologicalSpace.SeparableSpace α
-- TopologicalSpace.isSeparable_range
theorem isSeparable_range [TopologicalSpace β] [SeparableSpace α] {f : α → β} (hf : Continuous f) :
    IsSeparable (range f) 
Arguments: {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.SeparableSpace α] {f : α → β} (hf : Continuous f)
Goal: TopologicalSpace.IsSeparable (Set.range f)
-- TopologicalSpace.IsSeparable.of_subtype
theorem IsSeparable.of_subtype (s : Set α) [SeparableSpace s] : IsSeparable s 
Arguments: {α : Type u} [t : TopologicalSpace α] (s : Set α) [TopologicalSpace.SeparableSpace ↑s]
Goal: TopologicalSpace.IsSeparable s
-- TopologicalSpace.IsSeparable.of_separableSpace
theorem IsSeparable.of_separableSpace [h : SeparableSpace α] (s : Set α) : IsSeparable s 
Arguments: {α : Type u} [t : TopologicalSpace α] [h : TopologicalSpace.SeparableSpace α] (s : Set α)
Goal: TopologicalSpace.IsSeparable s
-- IsTopologicalBasis.iInf
protected theorem IsTopologicalBasis.iInf {β : Type*} {ι : Type*} {t : ι → TopologicalSpace β}
    {T : ι → Set (Set β)} (h_basis : ∀ i, IsTopologicalBasis (t := t i) (T i)) :
    IsTopologicalBasis (t := ⨅ i, t i)
      { S | ∃ (U : ι → Set β) (F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ i ∈ F, U i } 
Arguments: {β : Type u_1} {ι : Type u_2} {t : ι → TopologicalSpace β} {T : ι → Set (Set β)} (h_basis : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (T i))
Goal: TopologicalSpace.IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}
-- IsTopologicalBasis.iInf_induced
theorem IsTopologicalBasis.iInf_induced {β : Type*} {ι : Type*} {X : ι → Type*}
    [t : Π i, TopologicalSpace (X i)] {T : Π i, Set (Set (X i))}
    (cond : ∀ i, IsTopologicalBasis (T i)) (f : Π i, β → X i) :
    IsTopologicalBasis (t := ⨅ i, induced (f i) (t i))
      { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
        (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_ : i ∈ F), f i ⁻¹' U i } 
Arguments: {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)
Goal: TopologicalSpace.IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, f i ⁻¹' U i}
-- isTopologicalBasis_pi
theorem isTopologicalBasis_pi {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
    {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
    IsTopologicalBasis { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
      (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {T : (i : ι) → Set (Set (X i))} (cond : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (T i))
Goal: TopologicalSpace.IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = (↑F).pi U}
-- isTopologicalBasis_singletons
theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [DiscreteTopology α] :
    IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } 
Arguments: (α : Type u_1) [TopologicalSpace α] [DiscreteTopology α]
Goal: TopologicalSpace.IsTopologicalBasis {s | ∃ x, s = {x}}
-- isTopologicalBasis_subtype
theorem isTopologicalBasis_subtype
    {α : Type*} [TopologicalSpace α] {B : Set (Set α)}
    (h : TopologicalSpace.IsTopologicalBasis B) (p : α → Prop) :
    IsTopologicalBasis (Set.preimage (Subtype.val (p := p)) '' B) 
Arguments: {α : Type u_1} [TopologicalSpace α] {B : Set (Set α)} (h : TopologicalSpace.IsTopologicalBasis B) (p : α → Prop)
Goal: TopologicalSpace.IsTopologicalBasis (Set.preimage Subtype.val '' B)
-- isOpenMap_eval
lemma isOpenMap_eval (i : ι) : IsOpenMap (Function.eval i : (∀ i, π i) → π i) 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → TopologicalSpace (π i)] (i : ι)
Goal: IsOpenMap (Function.eval i)
-- Dense.exists_countable_dense_subset
theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {s : Set α}
    [SeparableSpace s] (hs : Dense s) : ∃ t ⊆ s, t.Countable ∧ Dense t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} [TopologicalSpace.SeparableSpace ↑s] (hs : Dense s)
Goal: ∃ t ⊆ s, t.Countable ∧ Dense t
-- Dense.exists_countable_dense_subset_bot_top
theorem Dense.exists_countable_dense_subset_bot_top {α : Type*} [TopologicalSpace α]
    [PartialOrder α] {s : Set α} [SeparableSpace s] (hs : Dense s) :
    ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧
      ∀ x, IsTop x → x ∈ s → x ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [PartialOrder α] {s : Set α} [TopologicalSpace.SeparableSpace ↑s] (hs : Dense s)
Goal: ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∈ s → x ∈ t) ∧ ∀ (x : α), IsTop x → x ∈ s → x ∈ t
-- exists_countable_dense_bot_top
theorem exists_countable_dense_bot_top (α : Type*) [TopologicalSpace α] [SeparableSpace α]
    [PartialOrder α] :
    ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∈ s) ∧ ∀ x, IsTop x → x ∈ s 
Arguments: (α : Type u_1) [TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [PartialOrder α]
Goal: ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∈ s) ∧ ∀ (x : α), IsTop x → x ∈ s
-- FirstCountableTopology.nhds_generated_countable
class _root_.FirstCountableTopology : Prop where
    nhds_generated_countable : ∀ a : α, (𝓝 a).IsCountablyGenerated
Arguments: {α : Type u} [t : TopologicalSpace α] [self : FirstCountableTopology α] (a : α)
Goal: (𝓝 a).IsCountablyGenerated
-- TopologicalSpace.firstCountableTopology_induced
theorem firstCountableTopology_induced (α β : Type*) [t : TopologicalSpace β]
    [FirstCountableTopology β] (f : α → β) : @FirstCountableTopology α (t.induced f) 
Arguments: (α : Type u_1) (β : Type u_2) [t : TopologicalSpace β] [FirstCountableTopology β] (f : α → β)
Goal: FirstCountableTopology α
-- Inducing.firstCountableTopology
protected theorem _root_.Inducing.firstCountableTopology {β : Type*}
    [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Inducing f) :
    FirstCountableTopology α 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_1} [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Inducing f)
Goal: FirstCountableTopology α
-- Embedding.firstCountableTopology
protected theorem _root_.Embedding.firstCountableTopology {β : Type*}
    [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Embedding f) :
    FirstCountableTopology α 
Arguments: {α : Type u} [t : TopologicalSpace α] {β : Type u_1} [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Embedding f)
Goal: FirstCountableTopology α
-- TopologicalSpace.FirstCountableTopology.tendsto_subseq
theorem tendsto_subseq [FirstCountableTopology α] {u : ℕ → α} {x : α}
    (hx : MapClusterPt x atTop u) : ∃ ψ : ℕ → ℕ, StrictMono ψ ∧ Tendsto (u ∘ ψ) atTop (𝓝 x) 
Arguments: {α : Type u} [t : TopologicalSpace α] [FirstCountableTopology α] {u : ℕ → α} {x : α} (hx : MapClusterPt x Filter.atTop u)
Goal: ∃ ψ, StrictMono ψ ∧ Filter.Tendsto (u ∘ ψ) Filter.atTop (𝓝 x)
-- SecondCountableTopology.is_open_generated_countable
class _root_.SecondCountableTopology : Prop where
    is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
Arguments: {α : Type u} [t : TopologicalSpace α] [self : SecondCountableTopology α]
Goal: ∃ b, b.Countable ∧ t = TopologicalSpace.generateFrom b