ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- interior_iInter₂_lt_nat
@[simp]
theorem interior_iInter₂_lt_nat {n : ℕ} (f : ℕ → Set X) :
    interior (⋂ m < n, f m) = ⋂ m < n, interior (f m) 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} (f : ℕ → Set X)
Goal: interior (⋂ m, ⋂ (_ : m < n), f m) = ⋂ m, ⋂ (_ : m < n), interior (f m)
-- interior_iInter₂_le_nat
@[simp]
theorem interior_iInter₂_le_nat {n : ℕ} (f : ℕ → Set X) :
    interior (⋂ m ≤ n, f m) = ⋂ m ≤ n, interior (f m) 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} (f : ℕ → Set X)
Goal: interior (⋂ m, ⋂ (_ : m ≤ n), f m) = ⋂ m, ⋂ (_ : m ≤ n), interior (f m)
-- interior_union_isClosed_of_interior_empty
theorem interior_union_isClosed_of_interior_empty (h₁ : IsClosed s)
    (h₂ : interior t = ∅) : interior (s ∪ t) = interior s 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : IsClosed s) (h₂ : interior t = ∅)
Goal: interior (s ∪ t) = interior s
-- isOpen_iff_forall_mem_open
theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ t ⊆ s, IsOpen t ∧ x ∈ t
-- interior_iInter_subset
theorem interior_iInter_subset (s : ι → Set X) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] (s : ι → Set X)
Goal: interior (⋂ i, s i) ⊆ ⋂ i, interior (s i)
-- interior_iInter₂_subset
theorem interior_iInter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set X) :
    interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] (p : ι → Sort u_3) (s : (i : ι) → p i → Set X)
Goal: interior (⋂ i, ⋂ j, s i j) ⊆ ⋂ i, ⋂ j, interior (s i j)
-- interior_sInter_subset
theorem interior_sInter_subset (S : Set (Set X)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s 
Arguments: {X : Type u} [TopologicalSpace X] (S : Set (Set X))
Goal: interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s
-- Filter.HasBasis.lift'_interior
theorem Filter.HasBasis.lift'_interior {l : Filter X} {p : ι → Prop} {s : ι → Set X}
    (h : l.HasBasis p s) : (l.lift' interior).HasBasis p fun i => interior (s i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {l : Filter X} {p : ι → Prop} {s : ι → Set X} (h : l.HasBasis p s)
Goal: (l.lift' interior).HasBasis p fun i => interior (s i)
-- Filter.lift'_interior_le
theorem Filter.lift'_interior_le (l : Filter X) : l.lift' interior ≤ l 
Arguments: {X : Type u} [TopologicalSpace X] (l : Filter X)
Goal: l.lift' interior ≤ l
-- Filter.HasBasis.lift'_interior_eq_self
theorem Filter.HasBasis.lift'_interior_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
    (h : l.HasBasis p s) (ho : ∀ i, p i → IsOpen (s i)) : l.lift' interior = l 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {l : Filter X} {p : ι → Prop} {s : ι → Set X} (h : l.HasBasis p s) (ho : ∀ (i : ι), p i → IsOpen (s i))
Goal: l.lift' interior = l
-- isClosed_closure
@[simp]
theorem isClosed_closure : IsClosed (closure s) 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed (closure s)
-- subset_closure
theorem subset_closure : s ⊆ closure s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: s ⊆ closure s
-- not_mem_of_not_mem_closure
theorem not_mem_of_not_mem_closure {P : X} (hP : P ∉ closure s) : P ∉ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] {P : X} (hP : P ∉ closure s)
Goal: P ∉ s
-- closure_minimal
theorem closure_minimal (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : s ⊆ t) (h₂ : IsClosed t)
Goal: closure s ⊆ t
-- Disjoint.closure_left
theorem Disjoint.closure_left (hd : Disjoint s t) (ht : IsOpen t) :
    Disjoint (closure s) t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hd : Disjoint s t) (ht : IsOpen t)
Goal: Disjoint (closure s) t
-- Disjoint.closure_right
theorem Disjoint.closure_right (hd : Disjoint s t) (hs : IsOpen s) :
    Disjoint s (closure t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hd : Disjoint s t) (hs : IsOpen s)
Goal: Disjoint s (closure t)
-- IsClosed.closure_eq
theorem IsClosed.closure_eq (h : IsClosed s) : closure s = s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (h : IsClosed s)
Goal: closure s = s
-- IsClosed.closure_subset
theorem IsClosed.closure_subset (hs : IsClosed s) : closure s ⊆ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : IsClosed s)
Goal: closure s ⊆ s
-- IsClosed.closure_subset_iff
theorem IsClosed.closure_subset_iff (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h₁ : IsClosed t)
Goal: closure s ⊆ t ↔ s ⊆ t
-- IsClosed.mem_iff_closure_subset
theorem IsClosed.mem_iff_closure_subset (hs : IsClosed s) :
    x ∈ s ↔ closure ({x} : Set X) ⊆ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (hs : IsClosed s)
Goal: x ∈ s ↔ closure {x} ⊆ s
-- closure_mono
@[mono, gcongr]
theorem closure_mono (h : s ⊆ t) : closure s ⊆ closure t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : s ⊆ t)
Goal: closure s ⊆ closure t
-- monotone_closure
theorem monotone_closure (X : Type*) [TopologicalSpace X] : Monotone (@closure X _) 
Arguments: (X : Type u_3) [TopologicalSpace X]
Goal: Monotone closure
-- diff_subset_closure_iff
theorem diff_subset_closure_iff : s \ t ⊆ closure t ↔ s ⊆ closure t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: s \ t ⊆ closure t ↔ s ⊆ closure t
-- closure_inter_subset_inter_closure
theorem closure_inter_subset_inter_closure (s t : Set X) :
    closure (s ∩ t) ⊆ closure s ∩ closure t 
Arguments: {X : Type u} [TopologicalSpace X] (s t : Set X)
Goal: closure (s ∩ t) ⊆ closure s ∩ closure t
-- isClosed_of_closure_subset
theorem isClosed_of_closure_subset (h : closure s ⊆ s) : IsClosed s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (h : closure s ⊆ s)
Goal: IsClosed s
-- closure_eq_iff_isClosed
theorem closure_eq_iff_isClosed : closure s = s ↔ IsClosed s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure s = s ↔ IsClosed s
-- closure_subset_iff_isClosed
theorem closure_subset_iff_isClosed : closure s ⊆ s ↔ IsClosed s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure s ⊆ s ↔ IsClosed s
-- closure_empty
@[simp]
theorem closure_empty : closure (∅ : Set X) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: closure ∅ = ∅
-- closure_empty_iff
@[simp]
theorem closure_empty_iff (s : Set X) : closure s = ∅ ↔ s = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: closure s = ∅ ↔ s = ∅
-- closure_nonempty_iff
@[simp]
theorem closure_nonempty_iff : (closure s).Nonempty ↔ s.Nonempty 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: (closure s).Nonempty ↔ s.Nonempty
-- closure_univ
@[simp]
theorem closure_univ : closure (univ : Set X) = univ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: closure Set.univ = Set.univ
-- closure_closure
@[simp]
theorem closure_closure : closure (closure s) = closure s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure (closure s) = closure s
-- closure_eq_compl_interior_compl
theorem closure_eq_compl_interior_compl : closure s = (interior sᶜ)ᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure s = (interior sᶜ)ᶜ
-- closure_union
@[simp]
theorem closure_union : closure (s ∪ t) = closure s ∪ closure t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: closure (s ∪ t) = closure s ∪ closure t
-- Set.Finite.closure_biUnion
theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} {s : Set ι} (hs : s.Finite) (f : ι → Set X)
Goal: closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i)
-- Set.Finite.closure_sUnion
theorem Set.Finite.closure_sUnion {S : Set (Set X)} (hS : S.Finite) :
    closure (⋃₀ S) = ⋃ s ∈ S, closure s 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite)
Goal: closure (⋃₀ S) = ⋃ s ∈ S, closure s
-- Finset.closure_biUnion
@[simp]
theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set X) :
    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} (s : Finset ι) (f : ι → Set X)
Goal: closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i)
-- closure_iUnion_of_finite
@[simp]
theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set X) :
    closure (⋃ i, f i) = ⋃ i, closure (f i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] [Finite ι] (f : ι → Set X)
Goal: closure (⋃ i, f i) = ⋃ i, closure (f i)
-- closure_iUnion₂_lt_nat
@[simp]
theorem closure_iUnion₂_lt_nat {n : ℕ} (f : ℕ → Set X) :
    closure (⋃ m < n, f m) = ⋃ m < n, closure (f m) 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} (f : ℕ → Set X)
Goal: closure (⋃ m, ⋃ (_ : m < n), f m) = ⋃ m, ⋃ (_ : m < n), closure (f m)
-- closure_iUnion₂_le_nat
@[simp]
theorem closure_iUnion₂_le_nat {n : ℕ} (f : ℕ → Set X) :
    closure (⋃ m ≤ n, f m) = ⋃ m ≤ n, closure (f m) 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} (f : ℕ → Set X)
Goal: closure (⋃ m, ⋃ (_ : m ≤ n), f m) = ⋃ m, ⋃ (_ : m ≤ n), closure (f m)
-- interior_subset_closure
theorem interior_subset_closure : interior s ⊆ closure s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s ⊆ closure s
-- interior_compl
@[simp]
theorem interior_compl : interior sᶜ = (closure s)ᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior sᶜ = (closure s)ᶜ
-- closure_compl
@[simp]
theorem closure_compl : closure sᶜ = (interior s)ᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure sᶜ = (interior s)ᶜ
-- mem_closure_iff
theorem mem_closure_iff :
    x ∈ closure s ↔ ∀ o, IsOpen o → x ∈ o → (o ∩ s).Nonempty 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ∀ (o : Set X), IsOpen o → x ∈ o → (o ∩ s).Nonempty
-- closure_inter_open_nonempty_iff
theorem closure_inter_open_nonempty_iff (h : IsOpen t) :
    (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : IsOpen t)
Goal: (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty
-- Filter.le_lift'_closure
theorem Filter.le_lift'_closure (l : Filter X) : l ≤ l.lift' closure 
Arguments: {X : Type u} [TopologicalSpace X] (l : Filter X)
Goal: l ≤ l.lift' closure
-- Filter.HasBasis.lift'_closure
theorem Filter.HasBasis.lift'_closure {l : Filter X} {p : ι → Prop} {s : ι → Set X}
    (h : l.HasBasis p s) : (l.lift' closure).HasBasis p fun i => closure (s i) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {l : Filter X} {p : ι → Prop} {s : ι → Set X} (h : l.HasBasis p s)
Goal: (l.lift' closure).HasBasis p fun i => closure (s i)
-- Filter.HasBasis.lift'_closure_eq_self
theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
    (h : l.HasBasis p s) (hc : ∀ i, p i → IsClosed (s i)) : l.lift' closure = l 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {l : Filter X} {p : ι → Prop} {s : ι → Set X} (h : l.HasBasis p s) (hc : ∀ (i : ι), p i → IsClosed (s i))
Goal: l.lift' closure = l
-- Filter.lift'_closure_eq_bot
@[simp]
theorem Filter.lift'_closure_eq_bot {l : Filter X} : l.lift' closure = ⊥ ↔ l = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] {l : Filter X}
Goal: l.lift' closure = ⊥ ↔ l = ⊥
-- dense_iff_closure_eq
theorem dense_iff_closure_eq : Dense s ↔ closure s = univ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: Dense s ↔ closure s = Set.univ
-- interior_eq_empty_iff_dense_compl
theorem interior_eq_empty_iff_dense_compl : interior s = ∅ ↔ Dense sᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s = ∅ ↔ Dense sᶜ
-- Dense.interior_compl
theorem Dense.interior_compl (h : Dense s) : interior sᶜ = ∅ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (h : Dense s)
Goal: interior sᶜ = ∅
-- dense_closure
@[simp]
theorem dense_closure : Dense (closure s) ↔ Dense s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: Dense (closure s) ↔ Dense s
-- dense_univ
@[simp]
theorem dense_univ : Dense (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Dense Set.univ
-- dense_iff_inter_open
theorem dense_iff_inter_open :
    Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: Dense s ↔ ∀ (U : Set X), IsOpen U → U.Nonempty → (U ∩ s).Nonempty
-- Dense.exists_mem_open
theorem Dense.exists_mem_open (hs : Dense s) {U : Set X} (ho : IsOpen U)
    (hne : U.Nonempty) : ∃ x ∈ s, x ∈ U 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : Dense s) {U : Set X} (ho : IsOpen U) (hne : U.Nonempty)
Goal: ∃ x ∈ s, x ∈ U
-- Dense.nonempty_iff
theorem Dense.nonempty_iff (hs : Dense s) : s.Nonempty ↔ Nonempty X 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : Dense s)
Goal: s.Nonempty ↔ Nonempty X
-- Dense.nonempty
theorem Dense.nonempty [h : Nonempty X] (hs : Dense s) : s.Nonempty 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] [h : Nonempty X] (hs : Dense s)
Goal: s.Nonempty
-- Dense.mono
@[mono]
theorem Dense.mono (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ 
Arguments: {X : Type u} {s₁ s₂ : Set X} [TopologicalSpace X] (h : s₁ ⊆ s₂) (hd : Dense s₁)
Goal: Dense s₂
-- dense_compl_singleton_iff_not_open
theorem dense_compl_singleton_iff_not_open :
    Dense ({x}ᶜ : Set X) ↔ ¬IsOpen ({x} : Set X) 
Arguments: {X : Type u} {x : X} [TopologicalSpace X]
Goal: Dense {x}ᶜ ↔ ¬IsOpen {x}
-- IsOpen.subset_interior_closure
theorem IsOpen.subset_interior_closure {s : Set X} (s_open : IsOpen s) :
    s ⊆ interior (closure s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (s_open : IsOpen s)
Goal: s ⊆ interior (closure s)
-- IsClosed.closure_interior_subset
theorem IsClosed.closure_interior_subset {s : Set X} (s_closed : IsClosed s) :
    closure (interior s) ⊆ s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (s_closed : IsClosed s)
Goal: closure (interior s) ⊆ s
-- closure_diff_interior
@[simp]
theorem closure_diff_interior (s : Set X) : closure s \ interior s = frontier s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: closure s \ interior s = frontier s
-- disjoint_interior_frontier
lemma disjoint_interior_frontier : Disjoint (interior s) (frontier s) 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: Disjoint (interior s) (frontier s)
-- closure_diff_frontier
@[simp]
theorem closure_diff_frontier (s : Set X) : closure s \ frontier s = interior s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: closure s \ frontier s = interior s
-- self_diff_frontier
@[simp]
theorem self_diff_frontier (s : Set X) : s \ frontier s = interior s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: s \ frontier s = interior s
-- frontier_eq_closure_inter_closure
theorem frontier_eq_closure_inter_closure : frontier s = closure s ∩ closure sᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier s = closure s ∩ closure sᶜ
-- frontier_subset_closure
theorem frontier_subset_closure : frontier s ⊆ closure s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier s ⊆ closure s
-- frontier_subset_iff_isClosed
theorem frontier_subset_iff_isClosed : frontier s ⊆ s ↔ IsClosed s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier s ⊆ s ↔ IsClosed s
-- frontier_closure_subset
theorem frontier_closure_subset : frontier (closure s) ⊆ frontier s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier (closure s) ⊆ frontier s
-- frontier_interior_subset
theorem frontier_interior_subset : frontier (interior s) ⊆ frontier s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier (interior s) ⊆ frontier s
-- frontier_compl
@[simp]
theorem frontier_compl (s : Set X) : frontier sᶜ = frontier s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: frontier sᶜ = frontier s
-- frontier_univ
@[simp]
theorem frontier_univ : frontier (univ : Set X) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: frontier Set.univ = ∅
-- frontier_empty
@[simp]
theorem frontier_empty : frontier (∅ : Set X) = ∅ 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: frontier ∅ = ∅
-- frontier_inter_subset
theorem frontier_inter_subset (s t : Set X) :
    frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t 
Arguments: {X : Type u} [TopologicalSpace X] (s t : Set X)
Goal: frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t
-- frontier_union_subset
theorem frontier_union_subset (s t : Set X) :
    frontier (s ∪ t) ⊆ frontier s ∩ closure tᶜ ∪ closure sᶜ ∩ frontier t 
Arguments: {X : Type u} [TopologicalSpace X] (s t : Set X)
Goal: frontier (s ∪ t) ⊆ frontier s ∩ closure tᶜ ∪ closure sᶜ ∩ frontier t
-- IsClosed.frontier_eq
theorem IsClosed.frontier_eq (hs : IsClosed s) : frontier s = s \ interior s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : IsClosed s)
Goal: frontier s = s \ interior s
-- IsOpen.frontier_eq
theorem IsOpen.frontier_eq (hs : IsOpen s) : frontier s = closure s \ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : IsOpen s)
Goal: frontier s = closure s \ s
-- IsOpen.inter_frontier_eq
theorem IsOpen.inter_frontier_eq (hs : IsOpen s) : s ∩ frontier s = ∅ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (hs : IsOpen s)
Goal: s ∩ frontier s = ∅
-- disjoint_frontier_iff_isOpen
theorem disjoint_frontier_iff_isOpen : Disjoint (frontier s) s ↔ IsOpen s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: Disjoint (frontier s) s ↔ IsOpen s
-- isClosed_frontier
theorem isClosed_frontier : IsClosed (frontier s) 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed (frontier s)
-- interior_frontier
theorem interior_frontier (h : IsClosed s) : interior (frontier s) = ∅ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] (h : IsClosed s)
Goal: interior (frontier s) = ∅
-- closure_eq_interior_union_frontier
theorem closure_eq_interior_union_frontier (s : Set X) : closure s = interior s ∪ frontier s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: closure s = interior s ∪ frontier s
-- closure_eq_self_union_frontier
theorem closure_eq_self_union_frontier (s : Set X) : closure s = s ∪ frontier s 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X)
Goal: closure s = s ∪ frontier s
-- Disjoint.frontier_left
theorem Disjoint.frontier_left (ht : IsOpen t) (hd : Disjoint s t) : Disjoint (frontier s) t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (ht : IsOpen t) (hd : Disjoint s t)
Goal: Disjoint (frontier s) t
-- Disjoint.frontier_right
theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s (frontier t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hs : IsOpen s) (hd : Disjoint s t)
Goal: Disjoint s (frontier t)
-- frontier_eq_inter_compl_interior
theorem frontier_eq_inter_compl_interior :
    frontier s = (interior s)ᶜ ∩ (interior sᶜ)ᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: frontier s = (interior s)ᶜ ∩ (interior sᶜ)ᶜ
-- compl_frontier_eq_union_interior
theorem compl_frontier_eq_union_interior :
    (frontier s)ᶜ = interior s ∪ interior sᶜ 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: (frontier s)ᶜ = interior s ∪ interior sᶜ
-- nhds_def'
theorem nhds_def' (x : X) : 𝓝 x = ⨅ (s : Set X) (_ : IsOpen s) (_ : x ∈ s), 𝓟 s 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: 𝓝 x = ⨅ s, ⨅ (_ : IsOpen s), ⨅ (_ : x ∈ s), 𝓟 s
-- nhds_basis_opens
theorem nhds_basis_opens (x : X) :
    (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsOpen s) fun s => s 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsOpen s) fun s => s
-- nhds_basis_closeds
theorem nhds_basis_closeds (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∉ s ∧ IsClosed s) compl 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => x ∉ s ∧ IsClosed s) compl
-- lift'_nhds_interior
@[simp]
theorem lift'_nhds_interior (x : X) : (𝓝 x).lift' interior = 𝓝 x 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: (𝓝 x).lift' interior = 𝓝 x
-- Filter.HasBasis.nhds_interior
theorem Filter.HasBasis.nhds_interior {x : X} {p : ι → Prop} {s : ι → Set X}
    (h : (𝓝 x).HasBasis p s) : (𝓝 x).HasBasis p (interior <| s ·) 
Arguments: {X : Type u} {ι : Sort w} [TopologicalSpace X] {x : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s)
Goal: (𝓝 x).HasBasis p fun x => interior (s x)
-- le_nhds_iff
theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Filter X}
Goal: f ≤ 𝓝 x ↔ ∀ (s : Set X), x ∈ s → IsOpen s → s ∈ f
-- nhds_le_of_le
theorem nhds_le_of_le {f} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 x ≤ f 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] {f : Filter X} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f)
Goal: 𝓝 x ≤ f
-- mem_nhds_iff
theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ t ⊆ s, IsOpen t ∧ x ∈ t 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: s ∈ 𝓝 x ↔ ∃ t ⊆ s, IsOpen t ∧ x ∈ t
-- eventually_nhds_iff
theorem eventually_nhds_iff {p : X → Prop} :
    (∀ᶠ y in 𝓝 x, p y) ↔ ∃ t : Set X, (∀ y ∈ t, p y) ∧ IsOpen t ∧ x ∈ t 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop}
Goal: (∀ᶠ (y : X) in 𝓝 x, p y) ↔ ∃ t, (∀ y ∈ t, p y) ∧ IsOpen t ∧ x ∈ t
-- frequently_nhds_iff
theorem frequently_nhds_iff {p : X → Prop} :
    (∃ᶠ y in 𝓝 x, p y) ↔ ∀ U : Set X, x ∈ U → IsOpen U → ∃ y ∈ U, p y 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop}
Goal: (∃ᶠ (y : X) in 𝓝 x, p y) ↔ ∀ (U : Set X), x ∈ U → IsOpen U → ∃ y ∈ U, p y
-- mem_interior_iff_mem_nhds
theorem mem_interior_iff_mem_nhds : x ∈ interior s ↔ s ∈ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ interior s ↔ s ∈ 𝓝 x
-- map_nhds
theorem map_nhds {f : X → α} :
    map f (𝓝 x) = ⨅ s ∈ { s : Set X | x ∈ s ∧ IsOpen s }, 𝓟 (f '' s) 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f : X → α}
Goal: Filter.map f (𝓝 x) = ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, 𝓟 (f '' s)
-- mem_of_mem_nhds
theorem mem_of_mem_nhds : s ∈ 𝓝 x → x ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: s ∈ 𝓝 x → x ∈ s
-- Filter.Eventually.self_of_nhds
theorem Filter.Eventually.self_of_nhds {p : X → Prop} (h : ∀ᶠ y in 𝓝 x, p y) : p x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop} (h : ∀ᶠ (y : X) in 𝓝 x, p y)
Goal: p x
-- IsOpen.mem_nhds
theorem IsOpen.mem_nhds (hs : IsOpen s) (hx : x ∈ s) : s ∈ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (hs : IsOpen s) (hx : x ∈ s)
Goal: s ∈ 𝓝 x
-- IsOpen.mem_nhds_iff
protected theorem IsOpen.mem_nhds_iff (hs : IsOpen s) : s ∈ 𝓝 x ↔ x ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (hs : IsOpen s)
Goal: s ∈ 𝓝 x ↔ x ∈ s
-- IsClosed.compl_mem_nhds
theorem IsClosed.compl_mem_nhds (hs : IsClosed s) (hx : x ∉ s) : sᶜ ∈ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (hs : IsClosed s) (hx : x ∉ s)
Goal: sᶜ ∈ 𝓝 x
-- IsOpen.eventually_mem
theorem IsOpen.eventually_mem (hs : IsOpen s) (hx : x ∈ s) :
    ∀ᶠ x in 𝓝 x, x ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (hs : IsOpen s) (hx : x ∈ s)
Goal: ∀ᶠ (x : X) in 𝓝 x, x ∈ s
-- nhds_basis_opens'
theorem nhds_basis_opens' (x : X) :
    (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsOpen s) fun x => x 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsOpen s) fun x => x
-- exists_open_set_nhds
theorem exists_open_set_nhds {U : Set X} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
    ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] {U : Set X} (h : ∀ x ∈ s, U ∈ 𝓝 x)
Goal: ∃ V, s ⊆ V ∧ IsOpen V ∧ V ⊆ U
-- exists_open_set_nhds'
theorem exists_open_set_nhds' {U : Set X} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
    ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] {U : Set X} (h : U ∈ ⨆ x ∈ s, 𝓝 x)
Goal: ∃ V, s ⊆ V ∧ IsOpen V ∧ V ⊆ U