ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAt.eventually_ne
theorem ContinuousAt.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y}
    (hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ z in 𝓝 x, g z ≠ y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y} (hg1 : ContinuousAt g x) (hg2 : g x ≠ y)
Goal: ∀ᶠ (z : X) in 𝓝 x, g z ≠ y
-- eventually_ne_nhds
theorem eventually_ne_nhds [T1Space X] {a b : X} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {a b : X} (h : a ≠ b)
Goal: ∀ᶠ (x : X) in 𝓝 a, x ≠ b
-- eventually_ne_nhdsWithin
theorem eventually_ne_nhdsWithin [T1Space X] {a b : X} {s : Set X} (h : a ≠ b) :
    ∀ᶠ x in 𝓝[s] a, x ≠ b 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {a b : X} {s : Set X} (h : a ≠ b)
Goal: ∀ᶠ (x : X) in 𝓝[s] a, x ≠ b
-- continuousAt_of_tendsto_nhds
theorem continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
    (h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x 
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: ContinuousAt f x
-- tendsto_const_nhds_iff
@[simp]
theorem tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} :
    Tendsto (fun _ => c) l (𝓝 d) ↔ c = d 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] {l : Filter Y} [l.NeBot] {c d : X}
Goal: Filter.Tendsto (fun x => c) l (𝓝 d) ↔ c = d
-- isOpen_singleton_of_finite_mem_nhds
theorem isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X)
    {s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (x : X) {s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite)
Goal: IsOpen {x}
-- infinite_of_mem_nhds
theorem infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)]
    {s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s 
Arguments: {X : Type u_3} [TopologicalSpace X] [T1Space X] (x : X) [hx : (𝓝[≠] x).NeBot] {s : Set X} (hs : s ∈ 𝓝 x)
Goal: s.Infinite
-- Set.Finite.continuousOn
theorem Set.Finite.continuousOn [T1Space X] [TopologicalSpace Y] {s : Set X} (hs : s.Finite)
    (f : X → Y) : ContinuousOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [TopologicalSpace Y] {s : Set X} (hs : s.Finite) (f : X → Y)
Goal: ContinuousOn f s
-- PreconnectedSpace.trivial_of_discrete
theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] :
    Subsingleton X 
Arguments: {X : Type u_1} [TopologicalSpace X] [PreconnectedSpace X] [DiscreteTopology X]
Goal: Subsingleton X
-- IsPreconnected.infinite_of_nontrivial
theorem IsPreconnected.infinite_of_nontrivial [T1Space X] {s : Set X} (h : IsPreconnected s)
    (hs : s.Nontrivial) : s.Infinite 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} (h : IsPreconnected s) (hs : s.Nontrivial)
Goal: s.Infinite
-- ConnectedSpace.infinite
theorem ConnectedSpace.infinite [ConnectedSpace X] [Nontrivial X] [T1Space X] : Infinite X 
Arguments: {X : Type u_1} [TopologicalSpace X] [ConnectedSpace X] [Nontrivial X] [T1Space X]
Goal: Infinite X
-- IsGδ.compl_singleton
theorem IsGδ.compl_singleton (x : X) [T1Space X] : IsGδ ({x}ᶜ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X) [T1Space X]
Goal: IsGδ {x}ᶜ
-- Set.Countable.isGδ_compl
theorem Set.Countable.isGδ_compl {s : Set X} [T1Space X] (hs : s.Countable) : IsGδ sᶜ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [T1Space X] (hs : s.Countable)
Goal: IsGδ sᶜ
-- Set.Finite.isGδ_compl
theorem Set.Finite.isGδ_compl {s : Set X} [T1Space X] (hs : s.Finite) : IsGδ sᶜ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [T1Space X] (hs : s.Finite)
Goal: IsGδ sᶜ
-- Set.Subsingleton.isGδ_compl
theorem Set.Subsingleton.isGδ_compl {s : Set X} [T1Space X] (hs : s.Subsingleton) : IsGδ sᶜ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [T1Space X] (hs : s.Subsingleton)
Goal: IsGδ sᶜ
-- Finset.isGδ_compl
theorem Finset.isGδ_compl [T1Space X] (s : Finset X) : IsGδ (sᶜ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (s : Finset X)
Goal: IsGδ (↑s)ᶜ
-- IsGδ.singleton
protected theorem IsGδ.singleton [T1Space X] (x : X) : IsGδ ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] [T1Space X] (x : X)
Goal: IsGδ {x}
-- Set.Finite.isGδ
theorem Set.Finite.isGδ {s : Set X} [T1Space X] (hs : s.Finite) : IsGδ s 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} [T1Space X] (hs : s.Finite)
Goal: IsGδ s
-- SeparationQuotient.t1Space_iff
theorem SeparationQuotient.t1Space_iff : T1Space (SeparationQuotient X) ↔ R0Space X 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T1Space (SeparationQuotient X) ↔ R0Space X
-- Set.Subsingleton.isClosed
lemma Set.Subsingleton.isClosed [T1Space X] {A : Set X} (h : A.Subsingleton) : IsClosed A 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {A : Set X} (h : A.Subsingleton)
Goal: IsClosed A
-- isClosed_inter_singleton
lemma isClosed_inter_singleton [T1Space X] {A : Set X} {a : X} : IsClosed (A ∩ {a}) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {A : Set X} {a : X}
Goal: IsClosed (A ∩ {a})
-- isClosed_singleton_inter
lemma isClosed_singleton_inter [T1Space X] {A : Set X} {a : X} : IsClosed ({a} ∩ A) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] {A : Set X} {a : X}
Goal: IsClosed ({a} ∩ A)
-- singleton_mem_nhdsWithin_of_mem_discrete
theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
    (hx : x ∈ s) : {x} ∈ 𝓝[s] x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [DiscreteTopology ↑s] {x : X} (hx : x ∈ s)
Goal: {x} ∈ 𝓝[s] x
-- nhdsWithin_of_mem_discrete
theorem nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
    𝓝[s] x = pure x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [DiscreteTopology ↑s] {x : X} (hx : x ∈ s)
Goal: 𝓝[s] x = pure x
-- Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete
theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop}
    {t : ι → Set X} {s : Set X} [DiscreteTopology s] {x : X} (hb : (𝓝 x).HasBasis p t)
    (hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Type u_3} {p : ι → Prop} {t : ι → Set X} {s : Set X} [DiscreteTopology ↑s] {x : X} (hb : (𝓝 x).HasBasis p t) (hx : x ∈ s)
Goal: ∃ i, p i ∧ t i ∩ s = {x}
-- nhds_inter_eq_singleton_of_mem_discrete
theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
    (hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [DiscreteTopology ↑s] {x : X} (hx : x ∈ s)
Goal: ∃ U ∈ 𝓝 x, U ∩ s = {x}
-- isOpen_inter_eq_singleton_of_mem_discrete
theorem isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
    (hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [DiscreteTopology ↑s] {x : X} (hx : x ∈ s)
Goal: ∃ U, IsOpen U ∧ U ∩ s = {x}
-- disjoint_nhdsWithin_of_mem_discrete
theorem disjoint_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
    ∃ U ∈ 𝓝[≠] x, Disjoint U s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [DiscreteTopology ↑s] {x : X} (hx : x ∈ s)
Goal: ∃ U ∈ 𝓝[≠] x, Disjoint U s
-- closedEmbedding_update
theorem closedEmbedding_update {ι : Type*} {β : ι → Type*}
    [DecidableEq ι] [(i : ι) → TopologicalSpace (β i)]
    (x : (i : ι) → β i) (i : ι) [(i : ι) → T1Space (β i)] :
    ClosedEmbedding (update x i) 
Arguments: {ι : Type u_3} {β : ι → Type u_4} [DecidableEq ι] [(i : ι) → TopologicalSpace (β i)] (x : (i : ι) → β i) (i : ι) [∀ (i : ι), T1Space (β i)]
Goal: ClosedEmbedding (Function.update x i)
-- R1Space.specializes_or_disjoint_nhds
@[mk_iff r1Space_iff_specializes_or_disjoint_nhds]
class R1Space (X : Type*) [TopologicalSpace X] : Prop where
  specializes_or_disjoint_nhds (x y : X) : Specializes x y ∨ Disjoint (𝓝 x) (𝓝 y)
Arguments: {X : Type u_3} [TopologicalSpace X] [self : R1Space X] (x y : X)
Goal: x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)
-- r1Space_iff_specializes_or_disjoint_nhds
@[mk_iff r1Space_iff_specializes_or_disjoint_nhds]
class R1Space (X : Type*) [TopologicalSpace X] : Prop where
  specializes_or_disjoint_nhds (x y : X) : Specializes x y ∨ Disjoint (𝓝 x) (𝓝 y)
Arguments: (X : Type u_3) [TopologicalSpace X]
Goal: R1Space X ↔ ∀ (x y : X), x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)
-- disjoint_nhds_nhds_iff_not_specializes
theorem disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {x y : X}
Goal: Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y
-- specializes_iff_not_disjoint
theorem specializes_iff_not_disjoint : x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {x y : X}
Goal: x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y)
-- disjoint_nhds_nhds_iff_not_inseparable
theorem disjoint_nhds_nhds_iff_not_inseparable : Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {x y : X}
Goal: Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y
-- r1Space_iff_inseparable_or_disjoint_nhds
theorem r1Space_iff_inseparable_or_disjoint_nhds {X : Type*} [TopologicalSpace X] :
    R1Space X ↔ ∀ x y : X, Inseparable x y ∨ Disjoint (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_3} [TopologicalSpace X]
Goal: R1Space X ↔ ∀ (x y : X), Inseparable x y ∨ Disjoint (𝓝 x) (𝓝 y)
-- isClosed_setOf_specializes
theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X]
Goal: IsClosed {p | p.1 ⤳ p.2}
-- isClosed_setOf_inseparable
theorem isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X]
Goal: IsClosed {p | Inseparable p.1 p.2}
-- IsCompact.mem_closure_iff_exists_inseparable
theorem IsCompact.mem_closure_iff_exists_inseparable {K : Set X} (hK : IsCompact K) :
    y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {y : X} {K : Set X} (hK : IsCompact K)
Goal: y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y
-- IsCompact.closure_eq_biUnion_inseparable
theorem IsCompact.closure_eq_biUnion_inseparable {K : Set X} (hK : IsCompact K) :
    closure K = ⋃ x ∈ K, {y | Inseparable x y} 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K : Set X} (hK : IsCompact K)
Goal: closure K = ⋃ x ∈ K, {y | Inseparable x y}
-- IsCompact.closure_eq_biUnion_closure_singleton
theorem IsCompact.closure_eq_biUnion_closure_singleton {K : Set X} (hK : IsCompact K) :
    closure K = ⋃ x ∈ K, closure {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K : Set X} (hK : IsCompact K)
Goal: closure K = ⋃ x ∈ K, closure {x}
-- IsCompact.closure_subset_of_isOpen
theorem IsCompact.closure_subset_of_isOpen {K : Set X} (hK : IsCompact K)
    {U : Set X} (hU : IsOpen U) (hKU : K ⊆ U) : closure K ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K : Set X} (hK : IsCompact K) {U : Set X} (hU : IsOpen U) (hKU : K ⊆ U)
Goal: closure K ⊆ U
-- IsCompact.closure
protected theorem IsCompact.closure {K : Set X} (hK : IsCompact K) : IsCompact (closure K) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K : Set X} (hK : IsCompact K)
Goal: IsCompact (closure K)
-- IsCompact.closure_of_subset
theorem IsCompact.closure_of_subset {s K : Set X} (hK : IsCompact K) (h : s ⊆ K) :
    IsCompact (closure s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {s K : Set X} (hK : IsCompact K) (h : s ⊆ K)
Goal: IsCompact (closure s)
-- exists_isCompact_superset_iff
@[simp]
theorem exists_isCompact_superset_iff {s : Set X} :
    (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {s : Set X}
Goal: (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s)
-- SeparatedNhds.of_isCompact_isCompact_isClosed
theorem SeparatedNhds.of_isCompact_isCompact_isClosed {K L : Set X} (hK : IsCompact K)
    (hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L) : SeparatedNhds K L 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K L : Set X} (hK : IsCompact K) (hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L)
Goal: SeparatedNhds K L
-- IsCompact.binary_compact_cover
theorem IsCompact.binary_compact_cover {K U V : Set X}
    (hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {K U V : Set X} (hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V)
Goal: ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂
-- IsCompact.finite_compact_cover
theorem IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type*}
    (t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {s : Set X} (hs : IsCompact s) {ι : Type u_3} (t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i)
Goal: ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i
-- R1Space.of_continuous_specializes_imp
theorem R1Space.of_continuous_specializes_imp [TopologicalSpace Y] {f : Y → X} (hc : Continuous f)
    (hspec : ∀ x y, f x ⤳ f y → x ⤳ y) : R1Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R1Space X] [TopologicalSpace Y] {f : Y → X} (hc : Continuous f) (hspec : ∀ (x y : Y), f x ⤳ f y → x ⤳ y)
Goal: R1Space Y
-- Inducing.r1Space
theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R1Space X] [TopologicalSpace Y] {f : Y → X} (hf : Inducing f)
Goal: R1Space Y
-- R1Space.induced
protected theorem R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [R1Space X] (f : Y → X)
Goal: R1Space Y
-- R1Space.sInf
protected theorem R1Space.sInf {X : Type*} {T : Set (TopologicalSpace X)}
    (hT : ∀ t ∈ T, @R1Space X t) : @R1Space X (sInf T) 
Arguments: {X : Type u_3} {T : Set (TopologicalSpace X)} (hT : ∀ t ∈ T, R1Space X)
Goal: R1Space X
-- R1Space.iInf
protected theorem R1Space.iInf {ι X : Type*} {t : ι → TopologicalSpace X}
    (ht : ∀ i, @R1Space X (t i)) : @R1Space X (iInf t) 
Arguments: {ι : Type u_3} {X : Type u_4} {t : ι → TopologicalSpace X} (ht : ∀ (i : ι), R1Space X)
Goal: R1Space X
-- R1Space.inf
protected theorem R1Space.inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
    (h₁ : @R1Space X t₁) (h₂ : @R1Space X t₂) : @R1Space X (t₁ ⊓ t₂) 
Arguments: {X : Type u_3} {t₁ t₂ : TopologicalSpace X} (h₁ : R1Space X) (h₂ : R1Space X)
Goal: R1Space X
-- exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
theorem exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
    {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [R1Space Y] {f : X → Y} {x : X}
    {K : Set X} {s : Set Y} (hf : Continuous f) (hs : s ∈ 𝓝 (f x)) (hKc : IsCompact K)
    (hKx : K ∈ 𝓝 x) : ∃ K ∈ 𝓝 x, IsCompact K ∧ MapsTo f K s 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [R1Space Y] {f : X → Y} {x : X} {K : Set X} {s : Set Y} (hf : Continuous f) (hs : s ∈ 𝓝 (f x)) (hKc : IsCompact K) (hKx : K ∈ 𝓝 x)
Goal: ∃ K ∈ 𝓝 x, IsCompact K ∧ Set.MapsTo f K s
-- IsCompact.isCompact_isClosed_basis_nhds
theorem IsCompact.isCompact_isClosed_basis_nhds {x : X} {L : Set X} (hLc : IsCompact L)
    (hxL : L ∈ 𝓝 x) : (𝓝 x).HasBasis (fun K ↦ K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] {x : X} {L : Set X} (hLc : IsCompact L) (hxL : L ∈ 𝓝 x)
Goal: (𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) fun x => x
-- Filter.coclosedCompact_eq_cocompact
@[simp]
theorem Filter.coclosedCompact_eq_cocompact : coclosedCompact X = cocompact X 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X]
Goal: Filter.coclosedCompact X = Filter.cocompact X
-- Bornology.relativelyCompact_eq_inCompact
@[simp]
theorem Bornology.relativelyCompact_eq_inCompact :
    Bornology.relativelyCompact X = Bornology.inCompact X 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X]
Goal: Bornology.relativelyCompact X = Bornology.inCompact X
-- isCompact_isClosed_basis_nhds
theorem isCompact_isClosed_basis_nhds (x : X) :
    (𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] [WeaklyLocallyCompactSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) fun x => x
-- exists_mem_nhds_isCompact_isClosed
theorem exists_mem_nhds_isCompact_isClosed (x : X) : ∃ K ∈ 𝓝 x, IsCompact K ∧ IsClosed K 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] [WeaklyLocallyCompactSpace X] (x : X)
Goal: ∃ K ∈ 𝓝 x, IsCompact K ∧ IsClosed K
-- exists_isOpen_superset_and_isCompact_closure
theorem exists_isOpen_superset_and_isCompact_closure {K : Set X} (hK : IsCompact K) :
    ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K)
Goal: ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V)
-- exists_isOpen_mem_isCompact_closure
theorem exists_isOpen_mem_isCompact_closure (x : X) :
    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X] [WeaklyLocallyCompactSpace X] (x : X)
Goal: ∃ U, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U)
-- t2Space_iff
@[mk_iff]
class T2Space (X : Type u) [TopologicalSpace X] : Prop where
    t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
Arguments: (X : Type u) [TopologicalSpace X]
Goal: T2Space X ↔ Pairwise fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
-- T2Space.t2
@[mk_iff]
class T2Space (X : Type u) [TopologicalSpace X] : Prop where
    t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
Arguments: {X : Type u} [TopologicalSpace X] [self : T2Space X]
Goal: Pairwise fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
-- t2_separation
theorem t2_separation [T2Space X] {x y : X} (h : x ≠ y) :
    ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x y : X} (h : x ≠ y)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
-- t2Space_iff_disjoint_nhds
theorem t2Space_iff_disjoint_nhds : T2Space X ↔ Pairwise fun x y : X => Disjoint (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space X ↔ Pairwise fun x y => Disjoint (𝓝 x) (𝓝 y)
-- disjoint_nhds_nhds
@[simp]
theorem disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x y : X}
Goal: Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y
-- pairwise_disjoint_nhds
theorem pairwise_disjoint_nhds [T2Space X] : Pairwise (Disjoint on (𝓝 : X → Filter X)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X]
Goal: Pairwise (Disjoint on 𝓝)
-- Set.pairwiseDisjoint_nhds
protected theorem Set.pairwiseDisjoint_nhds [T2Space X] (s : Set X) : s.PairwiseDisjoint 𝓝 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] (s : Set X)
Goal: s.PairwiseDisjoint 𝓝
-- Set.Finite.t2_separation
theorem Set.Finite.t2_separation [T2Space X] {s : Set X} (hs : s.Finite) :
    ∃ U : X → Set X, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {s : Set X} (hs : s.Finite)
Goal: ∃ U, (∀ (x : X), x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U
-- SeparationQuotient.t2Space_iff
theorem SeparationQuotient.t2Space_iff : T2Space (SeparationQuotient X) ↔ R1Space X 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space (SeparationQuotient X) ↔ R1Space X
-- R1Space.t2Space_iff_t0Space
theorem R1Space.t2Space_iff_t0Space [R1Space X] : T2Space X ↔ T0Space X 
Arguments: {X : Type u_1} [TopologicalSpace X] [R1Space X]
Goal: T2Space X ↔ T0Space X
-- t2_iff_nhds
theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space X ↔ ∀ {x y : X}, (𝓝 x ⊓ 𝓝 y).NeBot → x = y
-- eq_of_nhds_neBot
theorem eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x y : X} (h : (𝓝 x ⊓ 𝓝 y).NeBot)
Goal: x = y
-- t2Space_iff_nhds
theorem t2Space_iff_nhds :
    T2Space X ↔ Pairwise fun x y : X => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space X ↔ Pairwise fun x y => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V
-- t2_separation_nhds
theorem t2_separation_nhds [T2Space X] {x y : X} (h : x ≠ y) :
    ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x y : X} (h : x ≠ y)
Goal: ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v
-- t2_separation_compact_nhds
theorem t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
    ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y)
Goal: ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v
-- t2_iff_ultrafilter
theorem t2_iff_ultrafilter :
    T2Space X ↔ ∀ {x y : X} (f : Ultrafilter X), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space X ↔ ∀ {x y : X} (f : Ultrafilter X), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y
-- t2_iff_isClosed_diagonal
theorem t2_iff_isClosed_diagonal : T2Space X ↔ IsClosed (diagonal X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: T2Space X ↔ IsClosed (Set.diagonal X)
-- isClosed_diagonal
theorem isClosed_diagonal [T2Space X] : IsClosed (diagonal X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X]
Goal: IsClosed (Set.diagonal X)
-- tendsto_nhds_unique
theorem tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
    (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [l.NeBot] (ha : Filter.Tendsto f l (𝓝 a)) (hb : Filter.Tendsto f l (𝓝 b))
Goal: a = b
-- tendsto_nhds_unique'
theorem tendsto_nhds_unique' [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} (_ : NeBot l)
    (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {f : Y → X} {l : Filter Y} {a b : X}
Goal: l.NeBot → Filter.Tendsto f l (𝓝 a) → Filter.Tendsto f l (𝓝 b) → a = b
-- tendsto_nhds_unique_of_eventuallyEq
theorem tendsto_nhds_unique_of_eventuallyEq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
    [NeBot l] (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) : a = b 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X} [l.NeBot] (ha : Filter.Tendsto f l (𝓝 a)) (hb : Filter.Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g)
Goal: a = b
-- tendsto_nhds_unique_of_frequently_eq
theorem tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
    (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : ∃ᶠ x in l, f x = g x) : a = b 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X} (ha : Filter.Tendsto f l (𝓝 a)) (hb : Filter.Tendsto g l (𝓝 b)) (hfg : ∃ᶠ (x : Y) in l, f x = g x)
Goal: a = b
-- IsCompact.nhdsSet_inter_eq
theorem IsCompact.nhdsSet_inter_eq [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
    𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t)
Goal: 𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t