ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.Eventually.eventually_nhds
theorem Filter.Eventually.eventually_nhds {p : X → Prop} (h : ∀ᶠ y in 𝓝 x, p y) :
    ∀ᶠ y in 𝓝 x, ∀ᶠ x in 𝓝 y, p x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop} (h : ∀ᶠ (y : X) in 𝓝 x, p y)
Goal: ∀ᶠ (y : X) in 𝓝 x, ∀ᶠ (x : X) in 𝓝 y, p x
-- eventually_eventually_nhds
@[simp]
theorem eventually_eventually_nhds {p : X → Prop} :
    (∀ᶠ y in 𝓝 x, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝 x, p x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop}
Goal: (∀ᶠ (y : X) in 𝓝 x, ∀ᶠ (x : X) in 𝓝 y, p x) ↔ ∀ᶠ (x : X) in 𝓝 x, p x
-- frequently_frequently_nhds
@[simp]
theorem frequently_frequently_nhds {p : X → Prop} :
    (∃ᶠ x' in 𝓝 x, ∃ᶠ x'' in 𝓝 x', p x'') ↔ ∃ᶠ x in 𝓝 x, p x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {p : X → Prop}
Goal: (∃ᶠ (x' : X) in 𝓝 x, ∃ᶠ (x'' : X) in 𝓝 x', p x'') ↔ ∃ᶠ (x : X) in 𝓝 x, p x
-- eventually_mem_nhds
@[simp]
theorem eventually_mem_nhds : (∀ᶠ x' in 𝓝 x, s ∈ 𝓝 x') ↔ s ∈ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: (∀ᶠ (x' : X) in 𝓝 x, s ∈ 𝓝 x') ↔ s ∈ 𝓝 x
-- nhds_bind_nhds
@[simp]
theorem nhds_bind_nhds : (𝓝 x).bind 𝓝 = 𝓝 x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X]
Goal: (𝓝 x).bind 𝓝 = 𝓝 x
-- eventually_eventuallyEq_nhds
@[simp]
theorem eventually_eventuallyEq_nhds {f g : X → α} :
    (∀ᶠ y in 𝓝 x, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 x] g 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f g : X → α}
Goal: (∀ᶠ (y : X) in 𝓝 x, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 x] g
-- Filter.EventuallyEq.eq_of_nhds
theorem Filter.EventuallyEq.eq_of_nhds {f g : X → α} (h : f =ᶠ[𝓝 x] g) : f x = g x 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f g : X → α} (h : f =ᶠ[𝓝 x] g)
Goal: f x = g x
-- eventually_eventuallyLE_nhds
@[simp]
theorem eventually_eventuallyLE_nhds [LE α] {f g : X → α} :
    (∀ᶠ y in 𝓝 x, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 x] g 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] [LE α] {f g : X → α}
Goal: (∀ᶠ (y : X) in 𝓝 x, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 x] g
-- Filter.EventuallyEq.eventuallyEq_nhds
theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : X → α} (h : f =ᶠ[𝓝 x] g) :
    ∀ᶠ y in 𝓝 x, f =ᶠ[𝓝 y] g 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f g : X → α} (h : f =ᶠ[𝓝 x] g)
Goal: ∀ᶠ (y : X) in 𝓝 x, f =ᶠ[𝓝 y] g
-- Filter.EventuallyLE.eventuallyLE_nhds
theorem Filter.EventuallyLE.eventuallyLE_nhds [LE α] {f g : X → α} (h : f ≤ᶠ[𝓝 x] g) :
    ∀ᶠ y in 𝓝 x, f ≤ᶠ[𝓝 y] g 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] [LE α] {f g : X → α} (h : f ≤ᶠ[𝓝 x] g)
Goal: ∀ᶠ (y : X) in 𝓝 x, f ≤ᶠ[𝓝 y] g
-- all_mem_nhds
theorem all_mem_nhds (x : X) (P : Set X → Prop) (hP : ∀ s t, s ⊆ t → P s → P t) :
    (∀ s ∈ 𝓝 x, P s) ↔ ∀ s, IsOpen s → x ∈ s → P s 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (P : Set X → Prop) (hP : ∀ (s t : Set X), s ⊆ t → P s → P t)
Goal: (∀ s ∈ 𝓝 x, P s) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → P s
-- all_mem_nhds_filter
theorem all_mem_nhds_filter (x : X) (f : Set X → Set α) (hf : ∀ s t, s ⊆ t → f s ⊆ f t)
    (l : Filter α) : (∀ s ∈ 𝓝 x, f s ∈ l) ↔ ∀ s, IsOpen s → x ∈ s → f s ∈ l 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] (x : X) (f : Set X → Set α) (hf : ∀ (s t : Set X), s ⊆ t → f s ⊆ f t) (l : Filter α)
Goal: (∀ s ∈ 𝓝 x, f s ∈ l) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → f s ∈ l
-- tendsto_nhds
theorem tendsto_nhds {f : α → X} {l : Filter α} :
    Tendsto f l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → f ⁻¹' s ∈ l 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f : α → X} {l : Filter α}
Goal: Filter.Tendsto f l (𝓝 x) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → f ⁻¹' s ∈ l
-- tendsto_atTop_nhds
theorem tendsto_atTop_nhds [Nonempty α] [SemilatticeSup α] {f : α → X} :
    Tendsto f atTop (𝓝 x) ↔ ∀ U : Set X, x ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] [Nonempty α] [SemilatticeSup α] {f : α → X}
Goal: Filter.Tendsto f Filter.atTop (𝓝 x) ↔ ∀ (U : Set X), x ∈ U → IsOpen U → ∃ N, ∀ (n : α), N ≤ n → f n ∈ U
-- tendsto_const_nhds
theorem tendsto_const_nhds {f : Filter α} : Tendsto (fun _ : α => x) f (𝓝 x) 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f : Filter α}
Goal: Filter.Tendsto (fun x_1 => x) f (𝓝 x)
-- tendsto_atTop_of_eventually_const
theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι]
    {u : ι → X} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup ι] [Nonempty ι] {u : ι → X} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x)
Goal: Filter.Tendsto u Filter.atTop (𝓝 x)
-- tendsto_atBot_of_eventually_const
theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι]
    {u : ι → X} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf ι] [Nonempty ι] {u : ι → X} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x)
Goal: Filter.Tendsto u Filter.atBot (𝓝 x)
-- pure_le_nhds
theorem pure_le_nhds : pure ≤ (𝓝 : X → Filter X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: pure ≤ 𝓝
-- tendsto_pure_nhds
theorem tendsto_pure_nhds (f : α → X) (a : α) : Tendsto f (pure a) (𝓝 (f a)) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] (f : α → X) (a : α)
Goal: Filter.Tendsto f (pure a) (𝓝 (f a))
-- OrderTop.tendsto_atTop_nhds
theorem OrderTop.tendsto_atTop_nhds [PartialOrder α] [OrderTop α] (f : α → X) :
    Tendsto f atTop (𝓝 (f ⊤)) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] [PartialOrder α] [OrderTop α] (f : α → X)
Goal: Filter.Tendsto f Filter.atTop (𝓝 (f ⊤))
-- tendsto_nhds_of_eventually_eq
theorem tendsto_nhds_of_eventually_eq {l : Filter α} {f : α → X} (h : ∀ᶠ x' in l, f x' = x) :
    Tendsto f l (𝓝 x) 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {l : Filter α} {f : α → X} (h : ∀ᶠ (x' : α) in l, f x' = x)
Goal: Filter.Tendsto f l (𝓝 x)
-- Filter.EventuallyEq.tendsto
theorem Filter.EventuallyEq.tendsto {l : Filter α} {f : α → X} (hf : f =ᶠ[l] fun _ ↦ x) :
    Tendsto f l (𝓝 x) 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {l : Filter α} {f : α → X} (hf : f =ᶠ[l] fun x_1 => x)
Goal: Filter.Tendsto f l (𝓝 x)
-- ClusterPt.neBot
theorem ClusterPt.neBot {F : Filter X} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X} (h : ClusterPt x F)
Goal: (𝓝 x ⊓ F).NeBot
-- Filter.HasBasis.clusterPt_iff
theorem Filter.HasBasis.clusterPt_iff {ιX ιF} {pX : ιX → Prop} {sX : ιX → Set X} {pF : ιF → Prop}
    {sF : ιF → Set X} {F : Filter X} (hX : (𝓝 x).HasBasis pX sX) (hF : F.HasBasis pF sF) :
    ClusterPt x F ↔ ∀ ⦃i⦄, pX i → ∀ ⦃j⦄, pF j → (sX i ∩ sF j).Nonempty 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {ιX : Sort u_3} {ιF : Sort u_4} {pX : ιX → Prop} {sX : ιX → Set X} {pF : ιF → Prop} {sF : ιF → Set X} {F : Filter X} (hX : (𝓝 x).HasBasis pX sX) (hF : F.HasBasis pF sF)
Goal: ClusterPt x F ↔ ∀ ⦃i : ιX⦄, pX i → ∀ ⦃j : ιF⦄, pF j → (sX i ∩ sF j).Nonempty
-- clusterPt_iff
theorem clusterPt_iff {F : Filter X} :
    ClusterPt x F ↔ ∀ ⦃U : Set X⦄, U ∈ 𝓝 x → ∀ ⦃V⦄, V ∈ F → (U ∩ V).Nonempty 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x F ↔ ∀ ⦃U : Set X⦄, U ∈ 𝓝 x → ∀ ⦃V : Set X⦄, V ∈ F → (U ∩ V).Nonempty
-- clusterPt_iff_not_disjoint
theorem clusterPt_iff_not_disjoint {F : Filter X} :
    ClusterPt x F ↔ ¬Disjoint (𝓝 x) F 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x F ↔ ¬Disjoint (𝓝 x) F
-- clusterPt_principal_iff
theorem clusterPt_principal_iff :
    ClusterPt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).Nonempty 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: ClusterPt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).Nonempty
-- clusterPt_principal_iff_frequently
theorem clusterPt_principal_iff_frequently :
    ClusterPt x (𝓟 s) ↔ ∃ᶠ y in 𝓝 x, y ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: ClusterPt x (𝓟 s) ↔ ∃ᶠ (y : X) in 𝓝 x, y ∈ s
-- ClusterPt.of_le_nhds
theorem ClusterPt.of_le_nhds {f : Filter X} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Filter X} (H : f ≤ 𝓝 x) [f.NeBot]
Goal: ClusterPt x f
-- ClusterPt.of_le_nhds'
theorem ClusterPt.of_le_nhds' {f : Filter X} (H : f ≤ 𝓝 x) (_hf : NeBot f) :
    ClusterPt x f 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Filter X} (H : f ≤ 𝓝 x) (_hf : f.NeBot)
Goal: ClusterPt x f
-- ClusterPt.of_nhds_le
theorem ClusterPt.of_nhds_le {f : Filter X} (H : 𝓝 x ≤ f) : ClusterPt x f 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Filter X} (H : 𝓝 x ≤ f)
Goal: ClusterPt x f
-- ClusterPt.mono
theorem ClusterPt.mono {f g : Filter X} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f g : Filter X} (H : ClusterPt x f) (h : f ≤ g)
Goal: ClusterPt x g
-- ClusterPt.of_inf_left
theorem ClusterPt.of_inf_left {f g : Filter X} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f g : Filter X} (H : ClusterPt x (f ⊓ g))
Goal: ClusterPt x f
-- ClusterPt.of_inf_right
theorem ClusterPt.of_inf_right {f g : Filter X} (H : ClusterPt x <| f ⊓ g) :
    ClusterPt x g 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f g : Filter X} (H : ClusterPt x (f ⊓ g))
Goal: ClusterPt x g
-- Ultrafilter.clusterPt_iff
theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Ultrafilter X}
Goal: ClusterPt x ↑f ↔ ↑f ≤ 𝓝 x
-- clusterPt_iff_ultrafilter
theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {f : Filter X}
Goal: ClusterPt x f ↔ ∃ u, ↑u ≤ f ∧ ↑u ≤ 𝓝 x
-- mapClusterPt_def
theorem mapClusterPt_def {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
    MapClusterPt x F u ↔ ClusterPt x (map u F) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} (x : X) (F : Filter ι) (u : ι → X)
Goal: MapClusterPt x F u ↔ ClusterPt x (Filter.map u F)
-- mapClusterPt_iff
theorem mapClusterPt_iff {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
    MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} (x : X) (F : Filter ι) (u : ι → X)
Goal: MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ (a : ι) in F, u a ∈ s
-- mapClusterPt_iff_ultrafilter
theorem mapClusterPt_iff_ultrafilter {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter ι, U ≤ F ∧ Tendsto u U (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] {ι : Type u_3} (x : X) (F : Filter ι) (u : ι → X)
Goal: MapClusterPt x F u ↔ ∃ U, ↑U ≤ F ∧ Filter.Tendsto u (↑U) (𝓝 x)
-- mapClusterPt_comp
theorem mapClusterPt_comp {X α β : Type*} {x : X} [TopologicalSpace X] {F : Filter α} {φ : α → β}
    {u : β → X} : MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (map φ F) u 
Arguments: {X : Type u_3} {α : Type u_4} {β : Type u_5} {x : X} [TopologicalSpace X] {F : Filter α} {φ : α → β} {u : β → X}
Goal: MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (Filter.map φ F) u
-- mapClusterPt_of_comp
theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
    {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
    MapClusterPt x F u 
Arguments: {X : Type u} {α : Type u_1} {β : Type u_2} {x : X} [TopologicalSpace X] {F : Filter α} {φ : β → α} {p : Filter β} {u : α → X} [p.NeBot] (h : Filter.Tendsto φ p F) (H : Filter.Tendsto (u ∘ φ) p (𝓝 x))
Goal: MapClusterPt x F u
-- accPt_sup
theorem accPt_sup (x : X) (F G : Filter X) :
    AccPt x (F ⊔ G) ↔ AccPt x F ∨ AccPt x G 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (F G : Filter X)
Goal: AccPt x (F ⊔ G) ↔ AccPt x F ∨ AccPt x G
-- acc_iff_cluster
theorem acc_iff_cluster (x : X) (F : Filter X) : AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (F : Filter X)
Goal: AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F)
-- acc_principal_iff_cluster
theorem acc_principal_iff_cluster (x : X) (C : Set X) :
    AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (C : Set X)
Goal: AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x}))
-- accPt_iff_nhds
theorem accPt_iff_nhds (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (C : Set X)
Goal: AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x
-- accPt_iff_frequently
theorem accPt_iff_frequently (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (C : Set X)
Goal: AccPt x (𝓟 C) ↔ ∃ᶠ (y : X) in 𝓝 x, y ≠ x ∧ y ∈ C
-- AccPt.mono
theorem AccPt.mono {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G)
Goal: AccPt x G
-- AccPt.clusterPt
theorem AccPt.clusterPt (x : X) (F : Filter X) (h : AccPt x F) : ClusterPt x F 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) (F : Filter X) (h : AccPt x F)
Goal: ClusterPt x F
-- interior_eq_nhds'
theorem interior_eq_nhds' : interior s = { x | s ∈ 𝓝 x } 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s = {x | s ∈ 𝓝 x}
-- interior_eq_nhds
theorem interior_eq_nhds : interior s = { x | 𝓝 x ≤ 𝓟 s } 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: interior s = {x | 𝓝 x ≤ 𝓟 s}
-- interior_mem_nhds
@[simp]
theorem interior_mem_nhds : interior s ∈ 𝓝 x ↔ s ∈ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: interior s ∈ 𝓝 x ↔ s ∈ 𝓝 x
-- interior_setOf_eq
theorem interior_setOf_eq {p : X → Prop} : interior { x | p x } = { x | ∀ᶠ y in 𝓝 x, p y } 
Arguments: {X : Type u} [TopologicalSpace X] {p : X → Prop}
Goal: interior {x | p x} = {x | ∀ᶠ (y : X) in 𝓝 x, p y}
-- isOpen_setOf_eventually_nhds
theorem isOpen_setOf_eventually_nhds {p : X → Prop} : IsOpen { x | ∀ᶠ y in 𝓝 x, p y } 
Arguments: {X : Type u} [TopologicalSpace X] {p : X → Prop}
Goal: IsOpen {x | ∀ᶠ (y : X) in 𝓝 x, p y}
-- subset_interior_iff_nhds
theorem subset_interior_iff_nhds {V : Set X} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X] {V : Set X}
Goal: s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x
-- isOpen_iff_nhds
theorem isOpen_iff_nhds : IsOpen s ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen s ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s
-- TopologicalSpace.ext_iff_nhds
theorem TopologicalSpace.ext_iff_nhds {t t' : TopologicalSpace X} :
    t = t' ↔ ∀ x, @nhds _ t x = @nhds _ t' x 
Arguments: {X : Type u} {t t' : TopologicalSpace X}
Goal: t = t' ↔ ∀ (x : X), 𝓝 x = 𝓝 x
-- isOpen_iff_mem_nhds
theorem isOpen_iff_mem_nhds : IsOpen s ↔ ∀ x ∈ s, s ∈ 𝓝 x 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen s ↔ ∀ x ∈ s, s ∈ 𝓝 x
-- isOpen_iff_eventually
theorem isOpen_iff_eventually : IsOpen s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen s ↔ ∀ x ∈ s, ∀ᶠ (y : X) in 𝓝 x, y ∈ s
-- isOpen_iff_ultrafilter
theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l
-- isOpen_singleton_iff_nhds_eq_pure
theorem isOpen_singleton_iff_nhds_eq_pure (x : X) : IsOpen ({x} : Set X) ↔ 𝓝 x = pure x 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: IsOpen {x} ↔ 𝓝 x = pure x
-- isOpen_singleton_iff_punctured_nhds
theorem isOpen_singleton_iff_punctured_nhds (x : X) : IsOpen ({x} : Set X) ↔ 𝓝[≠] x = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: IsOpen {x} ↔ 𝓝[≠] x = ⊥
-- mem_closure_iff_frequently
theorem mem_closure_iff_frequently : x ∈ closure s ↔ ∃ᶠ x in 𝓝 x, x ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ∃ᶠ (x : X) in 𝓝 x, x ∈ s
-- isClosed_iff_frequently
theorem isClosed_iff_frequently : IsClosed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed s ↔ ∀ (x : X), (∃ᶠ (y : X) in 𝓝 x, y ∈ s) → x ∈ s
-- isClosed_setOf_clusterPt
theorem isClosed_setOf_clusterPt {f : Filter X} : IsClosed { x | ClusterPt x f } 
Arguments: {X : Type u} [TopologicalSpace X] {f : Filter X}
Goal: IsClosed {x | ClusterPt x f}
-- mem_closure_iff_clusterPt
theorem mem_closure_iff_clusterPt : x ∈ closure s ↔ ClusterPt x (𝓟 s) 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ClusterPt x (𝓟 s)
-- mem_closure_iff_nhds_ne_bot
theorem mem_closure_iff_nhds_ne_bot : x ∈ closure s ↔ 𝓝 x ⊓ 𝓟 s ≠ ⊥ 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ 𝓝 x ⊓ 𝓟 s ≠ ⊥
-- mem_closure_iff_nhdsWithin_neBot
theorem mem_closure_iff_nhdsWithin_neBot : x ∈ closure s ↔ NeBot (𝓝[s] x) 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ (𝓝[s] x).NeBot
-- nhdsWithin_neBot
lemma nhdsWithin_neBot : (𝓝[s] x).NeBot ↔ ∀ ⦃t⦄, t ∈ 𝓝 x → (t ∩ s).Nonempty 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: (𝓝[s] x).NeBot ↔ ∀ ⦃t : Set X⦄, t ∈ 𝓝 x → (t ∩ s).Nonempty
-- nhdsWithin_mono
@[gcongr]
theorem nhdsWithin_mono (x : X) {s t : Set X} (h : s ⊆ t) : 𝓝[s] x ≤ 𝓝[t] x 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) {s t : Set X} (h : s ⊆ t)
Goal: 𝓝[s] x ≤ 𝓝[t] x
-- not_mem_closure_iff_nhdsWithin_eq_bot
lemma not_mem_closure_iff_nhdsWithin_eq_bot : x ∉ closure s ↔ 𝓝[s] x = ⊥ 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∉ closure s ↔ 𝓝[s] x = ⊥
-- dense_compl_singleton
theorem dense_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) [(𝓝[≠] x).NeBot]
Goal: Dense {x}ᶜ
-- closure_compl_singleton
-- Porting note (#10618): was a `@[simp]` lemma but `simp` can prove it
theorem closure_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : closure {x}ᶜ = (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) [(𝓝[≠] x).NeBot]
Goal: closure {x}ᶜ = Set.univ
-- interior_singleton
@[simp]
theorem interior_singleton (x : X) [NeBot (𝓝[≠] x)] : interior {x} = (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) [(𝓝[≠] x).NeBot]
Goal: interior {x} = ∅
-- not_isOpen_singleton
theorem not_isOpen_singleton (x : X) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X) [(𝓝[≠] x).NeBot]
Goal: ¬IsOpen {x}
-- closure_eq_cluster_pts
theorem closure_eq_cluster_pts : closure s = { a | ClusterPt a (𝓟 s) } 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: closure s = {a | ClusterPt a (𝓟 s)}
-- mem_closure_iff_nhds
theorem mem_closure_iff_nhds : x ∈ closure s ↔ ∀ t ∈ 𝓝 x, (t ∩ s).Nonempty 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ∀ t ∈ 𝓝 x, (t ∩ s).Nonempty
-- mem_closure_iff_nhds'
theorem mem_closure_iff_nhds' : x ∈ closure s ↔ ∀ t ∈ 𝓝 x, ∃ y : s, ↑y ∈ t 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ∀ t ∈ 𝓝 x, ∃ y, ↑y ∈ t
-- mem_closure_iff_comap_neBot
theorem mem_closure_iff_comap_neBot :
    x ∈ closure s ↔ NeBot (comap ((↑) : s → X) (𝓝 x)) 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ (Filter.comap Subtype.val (𝓝 x)).NeBot
-- mem_closure_iff_nhds_basis'
theorem mem_closure_iff_nhds_basis' {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s) :
    x ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty 
Arguments: {X : Type u} {ι : Sort w} {x : X} {t : Set X} [TopologicalSpace X] {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s)
Goal: x ∈ closure t ↔ ∀ (i : ι), p i → (s i ∩ t).Nonempty
-- mem_closure_iff_nhds_basis
theorem mem_closure_iff_nhds_basis {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s) :
    x ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i 
Arguments: {X : Type u} {ι : Sort w} {x : X} {t : Set X} [TopologicalSpace X] {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s)
Goal: x ∈ closure t ↔ ∀ (i : ι), p i → ∃ y ∈ t, y ∈ s i
-- clusterPt_iff_forall_mem_closure
theorem clusterPt_iff_forall_mem_closure {F : Filter X} :
    ClusterPt x F ↔ ∀ s ∈ F, x ∈ closure s 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x F ↔ ∀ s ∈ F, x ∈ closure s
-- clusterPt_iff_lift'_closure
theorem clusterPt_iff_lift'_closure {F : Filter X} :
    ClusterPt x F ↔ pure x ≤ (F.lift' closure) 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x F ↔ pure x ≤ F.lift' closure
-- clusterPt_iff_lift'_closure'
theorem clusterPt_iff_lift'_closure' {F : Filter X} :
    ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot
-- clusterPt_lift'_closure_iff
@[simp]
theorem clusterPt_lift'_closure_iff {F : Filter X} :
    ClusterPt x (F.lift' closure) ↔ ClusterPt x F 
Arguments: {X : Type u} {x : X} [TopologicalSpace X] {F : Filter X}
Goal: ClusterPt x (F.lift' closure) ↔ ClusterPt x F
-- mem_closure_iff_ultrafilter
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X]
Goal: x ∈ closure s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x
-- isClosed_iff_clusterPt
theorem isClosed_iff_clusterPt : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed s ↔ ∀ (a : X), ClusterPt a (𝓟 s) → a ∈ s
-- isClosed_iff_ultrafilter
theorem isClosed_iff_ultrafilter : IsClosed s ↔
    ∀ x, ∀ u : Ultrafilter X, ↑u ≤ 𝓝 x → s ∈ u → x ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed s ↔ ∀ (x : X) (u : Ultrafilter X), ↑u ≤ 𝓝 x → s ∈ u → x ∈ s
-- isClosed_iff_nhds
theorem isClosed_iff_nhds :
    IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed s ↔ ∀ (x : X), (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s
-- isClosed_iff_forall_filter
lemma isClosed_iff_forall_filter :
    IsClosed s ↔ ∀ x, ∀ F : Filter X, F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s 
Arguments: {X : Type u} {s : Set X} [TopologicalSpace X]
Goal: IsClosed s ↔ ∀ (x : X) (F : Filter X), F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s
-- IsClosed.interior_union_left
theorem IsClosed.interior_union_left (_ : IsClosed s) :
    interior (s ∪ t) ⊆ s ∪ interior t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: IsClosed s → interior (s ∪ t) ⊆ s ∪ interior t
-- IsClosed.interior_union_right
theorem IsClosed.interior_union_right (h : IsClosed t) :
    interior (s ∪ t) ⊆ interior s ∪ t 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : IsClosed t)
Goal: interior (s ∪ t) ⊆ interior s ∪ t
-- IsOpen.inter_closure
theorem IsOpen.inter_closure (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : IsOpen s)
Goal: s ∩ closure t ⊆ closure (s ∩ t)
-- IsOpen.closure_inter
theorem IsOpen.closure_inter (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (h : IsOpen t)
Goal: closure s ∩ t ⊆ closure (s ∩ t)
-- Dense.open_subset_closure_inter
theorem Dense.open_subset_closure_inter (hs : Dense s) (ht : IsOpen t) :
    t ⊆ closure (t ∩ s) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hs : Dense s) (ht : IsOpen t)
Goal: t ⊆ closure (t ∩ s)
-- mem_closure_of_mem_closure_union
theorem mem_closure_of_mem_closure_union (h : x ∈ closure (s₁ ∪ s₂))
    (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ 
Arguments: {X : Type u} {x : X} {s₁ s₂ : Set X} [TopologicalSpace X] (h : x ∈ closure (s₁ ∪ s₂)) (h₁ : s₁ᶜ ∈ 𝓝 x)
Goal: x ∈ closure s₂
-- Dense.inter_of_isOpen_left
theorem Dense.inter_of_isOpen_left (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
    Dense (s ∩ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hs : Dense s) (ht : Dense t) (hso : IsOpen s)
Goal: Dense (s ∩ t)