ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- t0Space_iff_not_inseparable
theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: T0Space X ↔ Pairwise fun x y => ¬Inseparable x y
-- Inseparable.eq
theorem Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} (h : Inseparable x y)
Goal: x = y
-- Inducing.injective
protected theorem Inducing.injective [TopologicalSpace Y] [T0Space X] {f : X → Y}
    (hf : Inducing f) : Injective f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Inducing f)
Goal: Function.Injective f
-- Inducing.embedding
protected theorem Inducing.embedding [TopologicalSpace Y] [T0Space X] {f : X → Y}
    (hf : Inducing f) : Embedding f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Inducing f)
Goal: Embedding f
-- embedding_iff_inducing
lemma embedding_iff_inducing [TopologicalSpace Y] [T0Space X] {f : X → Y} :
    Embedding f ↔ Inducing f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] {f : X → Y}
Goal: Embedding f ↔ Inducing f
-- t0Space_iff_nhds_injective
theorem t0Space_iff_nhds_injective (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Injective (𝓝 : X → Filter X) 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: T0Space X ↔ Function.Injective 𝓝
-- nhds_injective
theorem nhds_injective [T0Space X] : Injective (𝓝 : X → Filter X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X]
Goal: Function.Injective 𝓝
-- inseparable_iff_eq
theorem inseparable_iff_eq [T0Space X] {x y : X} : Inseparable x y ↔ x = y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X}
Goal: Inseparable x y ↔ x = y
-- nhds_eq_nhds_iff
@[simp]
theorem nhds_eq_nhds_iff [T0Space X] {a b : X} : 𝓝 a = 𝓝 b ↔ a = b 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {a b : X}
Goal: 𝓝 a = 𝓝 b ↔ a = b
-- inseparable_eq_eq
@[simp]
theorem inseparable_eq_eq [T0Space X] : Inseparable = @Eq X 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X]
Goal: Inseparable = Eq
-- TopologicalSpace.IsTopologicalBasis.inseparable_iff
theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}
    (hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {b : Set (Set X)} (hb : TopologicalSpace.IsTopologicalBasis b) {x y : X}
Goal: Inseparable x y ↔ ∀ s ∈ b, x ∈ s ↔ y ∈ s
-- TopologicalSpace.IsTopologicalBasis.eq_iff
theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)}
    (hb : IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {b : Set (Set X)} (hb : TopologicalSpace.IsTopologicalBasis b) {x y : X}
Goal: x = y ↔ ∀ s ∈ b, x ∈ s ↔ y ∈ s
-- t0Space_iff_exists_isOpen_xor'_mem
theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: T0Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U)
-- exists_isOpen_xor'_mem
theorem exists_isOpen_xor'_mem [T0Space X] {x y : X} (h : x ≠ y) :
    ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {x y : X} (h : x ≠ y)
Goal: ∃ U, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U)
-- minimal_nonempty_closed_subsingleton
theorem minimal_nonempty_closed_subsingleton [T0Space X] {s : Set X} (hs : IsClosed s)
    (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : s.Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsClosed s) (hmin : ∀ t ⊆ s, t.Nonempty → IsClosed t → t = s)
Goal: s.Subsingleton
-- minimal_nonempty_closed_eq_singleton
theorem minimal_nonempty_closed_eq_singleton [T0Space X] {s : Set X} (hs : IsClosed s)
    (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} 
Arguments: {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)
Goal: ∃ x, s = {x}
-- IsClosed.exists_closed_singleton
theorem IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] [CompactSpace X] {S : Set X} (hS : IsClosed S) (hne : S.Nonempty)
Goal: ∃ x ∈ S, IsClosed {x}
-- minimal_nonempty_open_subsingleton
theorem minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s)
    (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : s.Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hs : IsOpen s) (hmin : ∀ t ⊆ s, t.Nonempty → IsOpen t → t = s)
Goal: s.Subsingleton
-- minimal_nonempty_open_eq_singleton
theorem minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen s)
    (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} 
Arguments: {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)
Goal: ∃ x, s = {x}
-- exists_isOpen_singleton_of_isOpen_finite
theorem exists_isOpen_singleton_of_isOpen_finite [T0Space X] {s : Set X} (hfin : s.Finite)
    (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] {s : Set X} (hfin : s.Finite) (hne : s.Nonempty) (ho : IsOpen s)
Goal: ∃ x ∈ s, IsOpen {x}
-- exists_open_singleton_of_finite
theorem exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] :
    ∃ x : X, IsOpen ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] [Finite X] [Nonempty X]
Goal: ∃ x, IsOpen {x}
-- t0Space_of_injective_of_continuous
theorem t0Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
    (hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Function.Injective f) (hf' : Continuous f) [T0Space Y]
Goal: T0Space X
-- Embedding.t0Space
protected theorem Embedding.t0Space [TopologicalSpace Y] [T0Space Y] {f : X → Y}
    (hf : Embedding f) : T0Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space Y] {f : X → Y} (hf : Embedding f)
Goal: T0Space X
-- t0Space_iff_or_not_mem_closure
theorem t0Space_iff_or_not_mem_closure (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: T0Space X ↔ Pairwise fun a b => a ∉ closure {b} ∨ b ∉ closure {a}
-- T0Space.of_cover
theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
    T0Space X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h : ∀ (x y : X), Inseparable x y → ∃ s, x ∈ s ∧ y ∈ s ∧ T0Space ↑s)
Goal: T0Space X
-- T0Space.of_open_cover
theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h : ∀ (x : X), ∃ s, x ∈ s ∧ IsOpen s ∧ T0Space ↑s)
Goal: T0Space X
-- R0Space.specializes_symmetric
@[mk_iff]
class R0Space (X : Type u) [TopologicalSpace X] : Prop where
    specializes_symmetric : Symmetric (Specializes : X → X → Prop)
Arguments: {X : Type u} [TopologicalSpace X] [self : R0Space X]
Goal: Symmetric Specializes
-- r0Space_iff
@[mk_iff]
class R0Space (X : Type u) [TopologicalSpace X] : Prop where
    specializes_symmetric : Symmetric (Specializes : X → X → Prop)
Arguments: (X : Type u) [TopologicalSpace X]
Goal: R0Space X ↔ Symmetric Specializes
-- Specializes.symm
theorem Specializes.symm (h : x ⤳ y) : y ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {x y : X} (h : x ⤳ y)
Goal: y ⤳ x
-- specializes_comm
theorem specializes_comm : x ⤳ y ↔ y ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {x y : X}
Goal: x ⤳ y ↔ y ⤳ x
-- specializes_iff_inseparable
theorem specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {x y : X}
Goal: x ⤳ y ↔ Inseparable x y
-- Inducing.r0Space
theorem Inducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R0Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R0Space X] [TopologicalSpace Y] {f : Y → X} (hf : Inducing f)
Goal: R0Space Y
-- isCompact_closure_singleton
theorem isCompact_closure_singleton : IsCompact (closure {x}) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {x : X}
Goal: IsCompact (closure {x})
-- Filter.coclosedCompact_le_cofinite
theorem Filter.coclosedCompact_le_cofinite : coclosedCompact X ≤ cofinite 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X]
Goal: Filter.coclosedCompact X ≤ Filter.cofinite
-- Bornology.relativelyCompact.isBounded_iff
theorem Bornology.relativelyCompact.isBounded_iff {s : Set X} :
    @Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {s : Set X}
Goal: Bornology.IsBounded s ↔ IsCompact (closure s)
-- Set.Finite.isCompact_closure
theorem Set.Finite.isCompact_closure {s : Set X} (hs : s.Finite) : IsCompact (closure s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R0Space X] {s : Set X} (hs : s.Finite)
Goal: IsCompact (closure s)
-- T1Space.t1
class T1Space (X : Type u) [TopologicalSpace X] : Prop where
    t1 : ∀ x, IsClosed ({x} : Set X)
Arguments: {X : Type u} [TopologicalSpace X] [self : T1Space X] (x : X)
Goal: IsClosed {x}
-- isClosed_singleton
theorem isClosed_singleton [T1Space X] {x : X} : IsClosed ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x : X}
Goal: IsClosed {x}
-- isOpen_compl_singleton
theorem isOpen_compl_singleton [T1Space X] {x : X} : IsOpen ({x}ᶜ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x : X}
Goal: IsOpen {x}ᶜ
-- isOpen_ne
theorem isOpen_ne [T1Space X] {x : X} : IsOpen { y | y ≠ x } 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x : X}
Goal: IsOpen {y | y ≠ x}
-- Continuous.isOpen_mulSupport
@[to_additive]
theorem Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X}
    (hf : Continuous f) : IsOpen (mulSupport f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X} (hf : Continuous f)
Goal: IsOpen (Function.mulSupport f)
-- Continuous.isOpen_support
@[to_additive]
theorem Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X}
    (hf : Continuous f) : IsOpen (mulSupport f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [Zero X] [TopologicalSpace Y] {f : Y → X} (hf : Continuous f)
Goal: IsOpen (Function.support f)
-- Ne.nhdsWithin_compl_singleton
theorem Ne.nhdsWithin_compl_singleton [T1Space X] {x y : X} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : x ≠ y)
Goal: 𝓝[{y}ᶜ] x = 𝓝 x
-- Ne.nhdsWithin_diff_singleton
theorem Ne.nhdsWithin_diff_singleton [T1Space X] {x y : X} (h : x ≠ y) (s : Set X) :
    𝓝[s \ {y}] x = 𝓝[s] x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : x ≠ y) (s : Set X)
Goal: 𝓝[s \ {y}] x = 𝓝[s] x
-- nhdsWithin_compl_singleton_le
lemma nhdsWithin_compl_singleton_le [T1Space X] (x y : X) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (x y : X)
Goal: 𝓝[≠] x ≤ 𝓝[{y}ᶜ] x
-- isOpen_setOf_eventually_nhdsWithin
theorem isOpen_setOf_eventually_nhdsWithin [T1Space X] {p : X → Prop} :
    IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {p : X → Prop}
Goal: IsOpen {x | ∀ᶠ (y : X) in 𝓝[≠] x, p y}
-- Set.Finite.isClosed
protected theorem Set.Finite.isClosed [T1Space X] {s : Set X} (hs : Set.Finite s) : IsClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} (hs : s.Finite)
Goal: IsClosed s
-- TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Set (Set X)}
    (hb : IsTopologicalBasis b) {x y : X} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {b : Set (Set X)} (hb : TopologicalSpace.IsTopologicalBasis b) {x y : X} (h : x ≠ y)
Goal: ∃ a ∈ b, x ∈ a ∧ y ∉ a
-- Finset.isClosed
protected theorem Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (s : Finset X)
Goal: IsClosed ↑s
-- t1Space_TFAE
theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
    List.TFAE [T1Space X,
      ∀ x, IsClosed ({ x } : Set X),
      ∀ x, IsOpen ({ x }ᶜ : Set X),
      Continuous (@CofiniteTopology.of X),
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
      ∀ ⦃x y : X⦄, x ⤳ y → x = y] 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
    ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
    ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
    ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE
-- t1Space_iff_continuous_cofinite_of
theorem t1Space_iff_continuous_cofinite_of : T1Space X ↔ Continuous (@CofiniteTopology.of X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space X ↔ Continuous ⇑CofiniteTopology.of
-- CofiniteTopology.continuous_of
theorem CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopology.of X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X]
Goal: Continuous ⇑CofiniteTopology.of
-- t1Space_iff_exists_open
theorem t1Space_iff_exists_open :
    T1Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
-- t1Space_iff_disjoint_pure_nhds
theorem t1Space_iff_disjoint_pure_nhds : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
-- t1Space_iff_disjoint_nhds_pure
theorem t1Space_iff_disjoint_nhds_pure : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
-- t1Space_iff_specializes_imp_eq
theorem t1Space_iff_specializes_imp_eq : T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y
-- disjoint_pure_nhds
theorem disjoint_pure_nhds [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : x ≠ y)
Goal: Disjoint (pure x) (𝓝 y)
-- disjoint_nhds_pure
theorem disjoint_nhds_pure [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : x ≠ y)
Goal: Disjoint (𝓝 x) (pure y)
-- Specializes.eq
theorem Specializes.eq [T1Space X] {x y : X} (h : x ⤳ y) : x = y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : x ⤳ y)
Goal: x = y
-- specializes_iff_eq
theorem specializes_iff_eq [T1Space X] {x y : X} : x ⤳ y ↔ x = y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X}
Goal: x ⤳ y ↔ x = y
-- specializes_eq_eq
@[simp] theorem specializes_eq_eq [T1Space X] : (· ⤳ ·) = @Eq X 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X]
Goal: (fun x x_1 => x ⤳ x_1) = Eq
-- pure_le_nhds_iff
@[simp]
theorem pure_le_nhds_iff [T1Space X] {a b : X} : pure a ≤ 𝓝 b ↔ a = b 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {a b : X}
Goal: pure a ≤ 𝓝 b ↔ a = b
-- nhds_le_nhds_iff
@[simp]
theorem nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {a b : X}
Goal: 𝓝 a ≤ 𝓝 b ↔ a = b
-- t1Space_antitone
theorem t1Space_antitone : Antitone (@T1Space X) 
Arguments: {X : Type u_1}
Goal: Antitone (@T1Space X)
-- continuousWithinAt_update_of_ne
theorem continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
    {s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x) :
    ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x' 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x)
Goal: ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x'
-- continuousAt_update_of_ne
theorem continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y]
    {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) :
    ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x)
Goal: ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x'
-- continuousOn_update_iff
theorem continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
    {s : Set X} {x : X} {y : Y} :
    ContinuousOn (Function.update f x y) s ↔
      ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x : X} {y : Y}
Goal: ContinuousOn (Function.update f x y) s ↔ ContinuousOn f (s \ {x}) ∧ (x ∈ s → Filter.Tendsto f (𝓝[s \ {x}] x) (𝓝 y))
-- t1Space_of_injective_of_continuous
theorem t1Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
    (hf : Function.Injective f) (hf' : Continuous f) [T1Space Y] : T1Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Function.Injective f) (hf' : Continuous f) [T1Space Y]
Goal: T1Space X
-- Embedding.t1Space
protected theorem Embedding.t1Space [TopologicalSpace Y] [T1Space Y] {f : X → Y}
    (hf : Embedding f) : T1Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {f : X → Y} (hf : Embedding f)
Goal: T1Space X
-- compl_singleton_mem_nhds_iff
@[simp]
theorem compl_singleton_mem_nhds_iff [T1Space X] {x y : X} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X}
Goal: {x}ᶜ ∈ 𝓝 y ↔ y ≠ x
-- compl_singleton_mem_nhds
theorem compl_singleton_mem_nhds [T1Space X] {x y : X} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} (h : y ≠ x)
Goal: {x}ᶜ ∈ 𝓝 y
-- closure_singleton
@[simp]
theorem closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x : X}
Goal: closure {x} = {x}
-- Set.Subsingleton.closure
theorem Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
    (closure s).Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} (hs : s.Subsingleton)
Goal: (closure s).Subsingleton
-- subsingleton_closure
@[simp]
theorem subsingleton_closure [T1Space X] {s : Set X} : (closure s).Subsingleton ↔ s.Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X}
Goal: (closure s).Subsingleton ↔ s.Subsingleton
-- isClosedMap_const
theorem isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} :
    IsClosedMap (Function.const X y) 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y}
Goal: IsClosedMap (Function.const X y)
-- nhdsWithin_insert_of_ne
theorem nhdsWithin_insert_of_ne [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y) :
    𝓝[insert y s] x = 𝓝[s] x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y)
Goal: 𝓝[insert y s] x = 𝓝[s] x
-- insert_mem_nhdsWithin_of_subset_insert
theorem insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X}
    (hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x y : X} {s t : Set X} (hu : t ⊆ insert y s)
Goal: insert x s ∈ 𝓝[t] x
-- ker_nhds
@[simp]
theorem ker_nhds [T1Space X] (x : X) : (𝓝 x).ker = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (x : X)
Goal: (𝓝 x).ker = {x}
-- biInter_basis_nhds
theorem biInter_basis_nhds [T1Space X] {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}
    (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {ι : Sort u_3} {p : ι → Prop} {s : ι → Set X} {x : X} (h : (𝓝 x).HasBasis p s)
Goal: ⋂ i, ⋂ (_ : p i), s i = {x}
-- compl_singleton_mem_nhdsSet_iff
@[simp]
theorem compl_singleton_mem_nhdsSet_iff [T1Space X] {x : X} {s : Set X} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {x : X} {s : Set X}
Goal: {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s
-- nhdsSet_le_iff
@[simp]
theorem nhdsSet_le_iff [T1Space X] {s t : Set X} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s t : Set X}
Goal: 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t
-- nhdsSet_inj_iff
@[simp]
theorem nhdsSet_inj_iff [T1Space X] {s t : Set X} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s t : Set X}
Goal: 𝓝ˢ s = 𝓝ˢ t ↔ s = t
-- injective_nhdsSet
theorem injective_nhdsSet [T1Space X] : Function.Injective (𝓝ˢ : Set X → Filter X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X]
Goal: Function.Injective 𝓝ˢ
-- strictMono_nhdsSet
theorem strictMono_nhdsSet [T1Space X] : StrictMono (𝓝ˢ : Set X → Filter X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X]
Goal: StrictMono 𝓝ˢ
-- nhds_le_nhdsSet_iff
@[simp]
theorem nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} {x : X}
Goal: 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s
-- Dense.diff_singleton
theorem Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] :
    Dense (s \ {x}) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} (hs : Dense s) (x : X) [(𝓝[≠] x).NeBot]
Goal: Dense (s \ {x})
-- Dense.diff_finset
theorem Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
    (t : Finset X) : Dense (s \ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] [∀ (x : X), (𝓝[≠] x).NeBot] {s : Set X} (hs : Dense s) (t : Finset X)
Goal: Dense (s \ ↑t)
-- Dense.diff_finite
theorem Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
    {t : Set X} (ht : t.Finite) : Dense (s \ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] [∀ (x : X), (𝓝[≠] x).NeBot] {s : Set X} (hs : Dense s) {t : Set X} (ht : t.Finite)
Goal: Dense (s \ t)
-- eq_of_tendsto_nhds
theorem eq_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
    (h : Tendsto f (𝓝 x) (𝓝 y)) : f x = y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y} (h : Filter.Tendsto f (𝓝 x) (𝓝 y))
Goal: f x = y
-- Filter.Tendsto.eventually_ne
theorem Filter.Tendsto.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y}
    {l : Filter X} {b₁ b₂ : Y} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] [T1Space Y] {g : X → Y} {l : Filter X} {b₁ b₂ : Y} (hg : Filter.Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂)
Goal: ∀ᶠ (z : X) in l, g z ≠ b₂