ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.InjOn.exists_mem_nhdsSet
theorem Set.InjOn.exists_mem_nhdsSet {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    [T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
    (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
    ∃ t ∈ 𝓝ˢ s, InjOn f t 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} {s : Set X} (inj : Set.InjOn f s) (sc : IsCompact s) (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, Set.InjOn f u)
Goal: ∃ t ∈ 𝓝ˢ s, Set.InjOn f t
-- Set.InjOn.exists_isOpen_superset
theorem Set.InjOn.exists_isOpen_superset {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    [T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
    (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
    ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn f t 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} {s : Set X} (inj : Set.InjOn f s) (sc : IsCompact s) (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, Set.InjOn f u)
Goal: ∃ t, IsOpen t ∧ s ⊆ t ∧ Set.InjOn f t
-- lim_eq
theorem lim_eq {x : X} [NeBot f] (h : f ≤ 𝓝 x) : @lim _ _ ⟨x⟩ f = x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {f : Filter X} {x : X} [f.NeBot] (h : f ≤ 𝓝 x)
Goal: lim f = x
-- lim_eq_iff
theorem lim_eq_iff [NeBot f] (h : ∃ x : X, f ≤ 𝓝 x) {x} : @lim _ _ ⟨x⟩ f = x ↔ f ≤ 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {f : Filter X} [f.NeBot] (h : ∃ x, f ≤ 𝓝 x) {x : X}
Goal: lim f = x ↔ f ≤ 𝓝 x
-- Ultrafilter.lim_eq_iff_le_nhds
theorem Ultrafilter.lim_eq_iff_le_nhds [CompactSpace X] {x : X} {F : Ultrafilter X} :
    F.lim = x ↔ ↑F ≤ 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] {x : X} {F : Ultrafilter X}
Goal: F.lim = x ↔ ↑F ≤ 𝓝 x
-- isOpen_iff_ultrafilter'
theorem isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
    IsOpen U ↔ ∀ F : Ultrafilter X, F.lim ∈ U → U ∈ F.1 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] (U : Set X)
Goal: IsOpen U ↔ ∀ (F : Ultrafilter X), F.lim ∈ U → U ∈ ↑F
-- Filter.Tendsto.limUnder_eq
theorem Filter.Tendsto.limUnder_eq {x : X} {f : Filter Y} [NeBot f] {g : Y → X}
    (h : Tendsto g f (𝓝 x)) : @limUnder _ _ _ ⟨x⟩ f g = x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {x : X} {f : Filter Y} [f.NeBot] {g : Y → X} (h : Filter.Tendsto g f (𝓝 x))
Goal: limUnder f g = x
-- Filter.limUnder_eq_iff
theorem Filter.limUnder_eq_iff {f : Filter Y} [NeBot f] {g : Y → X} (h : ∃ x, Tendsto g f (𝓝 x))
    {x} : @limUnder _ _ _ ⟨x⟩ f g = x ↔ Tendsto g f (𝓝 x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] {f : Filter Y} [f.NeBot] {g : Y → X} (h : ∃ x, Filter.Tendsto g f (𝓝 x)) {x : X}
Goal: limUnder f g = x ↔ Filter.Tendsto g f (𝓝 x)
-- Continuous.limUnder_eq
theorem Continuous.limUnder_eq [TopologicalSpace Y] {f : Y → X} (h : Continuous f) (y : Y) :
    @limUnder _ _ _ ⟨f y⟩ (𝓝 y) f = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T2Space X] [TopologicalSpace Y] {f : Y → X} (h : Continuous f) (y : Y)
Goal: limUnder (𝓝 y) f = f y
-- lim_nhds
@[simp]
theorem lim_nhds (x : X) : @lim _ _ ⟨x⟩ (𝓝 x) = x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] (x : X)
Goal: lim (𝓝 x) = x
-- limUnder_nhds_id
@[simp]
theorem limUnder_nhds_id (x : X) : @limUnder _ _ _ ⟨x⟩ (𝓝 x) id = x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] (x : X)
Goal: limUnder (𝓝 x) id = x
-- lim_nhdsWithin
@[simp]
theorem lim_nhdsWithin {x : X} {s : Set X} (h : x ∈ closure s) : @lim _ _ ⟨x⟩ (𝓝[s] x) = x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X} {s : Set X} (h : x ∈ closure s)
Goal: lim (𝓝[s] x) = x
-- limUnder_nhdsWithin_id
@[simp]
theorem limUnder_nhdsWithin_id {x : X} {s : Set X} (h : x ∈ closure s) :
    @limUnder _ _ _ ⟨x⟩ (𝓝[s] x) id = x 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X} {s : Set X} (h : x ∈ closure s)
Goal: limUnder (𝓝[s] x) id = x
-- separated_by_continuous
theorem separated_by_continuous [TopologicalSpace Y] [T2Space Y]
    {f : X → Y} (hf : Continuous f) {x y : X} (h : f x ≠ f y) :
    ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Continuous f) {x y : X} (h : f x ≠ f y)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
-- separated_by_openEmbedding
theorem separated_by_openEmbedding [TopologicalSpace Y] [T2Space X]
    {f : X → Y} (hf : OpenEmbedding f) {x y : X} (h : x ≠ y) :
    ∃ u v : Set Y, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {f : X → Y} (hf : OpenEmbedding f) {x y : X} (h : x ≠ y)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v
-- T2Space.of_injective_continuous
theorem T2Space.of_injective_continuous [TopologicalSpace Y] [T2Space Y] {f : X → Y}
    (hinj : Injective f) (hc : Continuous f) : T2Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hinj : Function.Injective f) (hc : Continuous f)
Goal: T2Space X
-- Embedding.t2Space
theorem Embedding.t2Space [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Embedding f) :
    T2Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Embedding f)
Goal: T2Space X
-- t2Quotient.mk_eq
lemma mk_eq {x y : X} : mk x = mk y ↔ ∀ s : Setoid X, T2Space (Quotient s) → s.Rel x y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: t2Quotient.mk x = t2Quotient.mk y ↔ ∀ (s : Setoid X), T2Space (Quotient s) → s.Rel x y
-- t2Quotient.surjective_mk
lemma surjective_mk : Surjective (mk : X → t2Quotient X) 
Arguments: (X : Type u_1) [TopologicalSpace X]
Goal: Function.Surjective t2Quotient.mk
-- t2Quotient.continuous_mk
lemma continuous_mk : Continuous (mk : X → t2Quotient X) 
Arguments: (X : Type u_1) [TopologicalSpace X]
Goal: Continuous t2Quotient.mk
-- t2Quotient.inductionOn
@[elab_as_elim]
protected lemma inductionOn {motive : t2Quotient X → Prop} (q : t2Quotient X)
    (h : ∀ x, motive (t2Quotient.mk x)) : motive q 
Arguments: {X : Type u_1} [TopologicalSpace X] {motive : t2Quotient X → Prop} (q : t2Quotient X) (h : ∀ (x : X), motive (t2Quotient.mk x))
Goal: motive q
-- t2Quotient.inductionOn₂
@[elab_as_elim]
protected lemma inductionOn₂ [TopologicalSpace Y] {motive : t2Quotient X → t2Quotient Y → Prop}
    (q : t2Quotient X) (q' : t2Quotient Y) (h : ∀ x y, motive (mk x) (mk y)) : motive q q' 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {motive : t2Quotient X → t2Quotient Y → Prop} (q : t2Quotient X) (q' : t2Quotient Y) (h : ∀ (x : X) (y : Y), motive (t2Quotient.mk x) (t2Quotient.mk y))
Goal: motive q q'
-- t2Quotient.compatible
lemma compatible {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
    {f : X → Y} (hf : Continuous f) : letI _ := t2Setoid X
    ∀ (a b : X), a ≈ b → f a = f b 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Continuous f) (a b : X)
Goal: a ≈ b → f a = f b
-- t2Quotient.continuous_lift
lemma continuous_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
    {f : X → Y} (hf : Continuous f) : Continuous (t2Quotient.lift hf) 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Continuous f)
Goal: Continuous (t2Quotient.lift hf)
-- t2Quotient.lift_mk
@[simp]
lemma lift_mk {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
    {f : X → Y} (hf : Continuous f) (x : X) : lift hf (mk x) = f x 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Continuous f) (x : X)
Goal: t2Quotient.lift hf (t2Quotient.mk x) = f x
-- t2Quotient.unique_lift
lemma unique_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
    {f : X → Y} (hf : Continuous f) {g : t2Quotient X → Y} (hfg : g ∘ mk = f) :
    g = lift hf 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Continuous f) {g : t2Quotient X → Y} (hfg : g ∘ t2Quotient.mk = f)
Goal: g = t2Quotient.lift hf
-- isClosed_eq
theorem isClosed_eq [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
    IsClosed { y : Y | f y = g y } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g)
Goal: IsClosed {y | f y = g y}
-- isOpen_ne_fun
theorem isOpen_ne_fun [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
    IsOpen { y : Y | f y ≠ g y } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g)
Goal: IsOpen {y | f y ≠ g y}
-- Set.EqOn.closure
protected theorem Set.EqOn.closure [T2Space X] {s : Set Y} {f g : Y → X} (h : EqOn f g s)
    (hf : Continuous f) (hg : Continuous g) : EqOn f g (closure s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {s : Set Y} {f g : Y → X} (h : Set.EqOn f g s) (hf : Continuous f) (hg : Continuous g)
Goal: Set.EqOn f g (closure s)
-- Continuous.ext_on
theorem Continuous.ext_on [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f)
    (hg : Continuous g) (h : EqOn f g s) : f = g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : Set.EqOn f g s)
Goal: f = g
-- eqOn_closure₂'
theorem eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
    (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
    (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
    (hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_3} [TopologicalSpace Y] [TopologicalSpace Z] [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z} (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ (x : X), Continuous (f x)) (hf₂ : ∀ (y : Y), Continuous fun x => f x y) (hg₁ : ∀ (x : X), Continuous (g x)) (hg₂ : ∀ (y : Y), Continuous fun x => g x y) (x : X)
Goal: x ∈ closure s → ∀ y ∈ closure t, f x y = g x y
-- eqOn_closure₂
theorem eqOn_closure₂ [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
    (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
    (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_3} [TopologicalSpace Y] [TopologicalSpace Z] [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z} (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (Function.uncurry f)) (hg : Continuous (Function.uncurry g)) (x : X)
Goal: x ∈ closure s → ∀ y ∈ closure t, f x y = g x y
-- Set.EqOn.of_subset_closure
theorem Set.EqOn.of_subset_closure [T2Space Y] {s t : Set X} {f g : X → Y} (h : EqOn f g s)
    (hf : ContinuousOn f t) (hg : ContinuousOn g t) (hst : s ⊆ t) (hts : t ⊆ closure s) :
    EqOn f g t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {s t : Set X} {f g : X → Y} (h : Set.EqOn f g s) (hf : ContinuousOn f t) (hg : ContinuousOn g t) (hst : s ⊆ t) (hts : t ⊆ closure s)
Goal: Set.EqOn f g t
-- Function.LeftInverse.isClosed_range
theorem Function.LeftInverse.isClosed_range [T2Space X] {f : X → Y} {g : Y → X}
    (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {f : X → Y} {g : Y → X} (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g)
Goal: IsClosed (Set.range g)
-- Function.LeftInverse.closedEmbedding
theorem Function.LeftInverse.closedEmbedding [T2Space X] {f : X → Y} {g : Y → X}
    (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : ClosedEmbedding g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {f : X → Y} {g : Y → X} (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g)
Goal: ClosedEmbedding g
-- SeparatedNhds.of_isCompact_isCompact
theorem SeparatedNhds.of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)
    (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) (hst : Disjoint s t)
Goal: SeparatedNhds s t
-- SeparatedNhds.of_finset_finset
theorem SeparatedNhds.of_finset_finset [T2Space X] (s t : Finset X) (h : Disjoint s t) :
    SeparatedNhds (s : Set X) t 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] (s t : Finset X) (h : Disjoint s t)
Goal: SeparatedNhds ↑s ↑t
-- SeparatedNhds.of_singleton_finset
theorem SeparatedNhds.of_singleton_finset [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
    SeparatedNhds ({x} : Set X) s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X} {s : Finset X} (h : x ∉ s)
Goal: SeparatedNhds {x} ↑s
-- IsCompact.isClosed
theorem IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {s : Set X} (hs : IsCompact s)
Goal: IsClosed s
-- IsCompact.preimage_continuous
theorem IsCompact.preimage_continuous [CompactSpace X] [T2Space Y] {f : X → Y} {s : Set Y}
    (hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] [T2Space Y] {f : X → Y} {s : Set Y} (hs : IsCompact s) (hf : Continuous f)
Goal: IsCompact (f ⁻¹' s)
-- Pi.isCompact_iff
lemma Pi.isCompact_iff {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    [∀ i, T2Space (π i)] {s : Set (Π i, π i)} :
    IsCompact s ↔ IsClosed s ∧ ∀ i, IsCompact (eval i '' s) 
Arguments: {ι : Type u_4} {π : ι → Type u_5} [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), T2Space (π i)] {s : Set ((i : ι) → π i)}
Goal: IsCompact s ↔ IsClosed s ∧ ∀ (i : ι), IsCompact (Function.eval i '' s)
-- Pi.isCompact_closure_iff
lemma Pi.isCompact_closure_iff {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    [∀ i, T2Space (π i)] {s : Set (Π i, π i)} :
    IsCompact (closure s) ↔ ∀ i, IsCompact (closure <| eval i '' s) 
Arguments: {ι : Type u_4} {π : ι → Type u_5} [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), T2Space (π i)] {s : Set ((i : ι) → π i)}
Goal: IsCompact (closure s) ↔ ∀ (i : ι), IsCompact (closure (Function.eval i '' s))
-- exists_subset_nhds_of_isCompact
theorem exists_subset_nhds_of_isCompact [T2Space X] {ι : Type*} [Nonempty ι] {V : ι → Set X}
    (hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) {U : Set X}
    (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {ι : Type u_4} [Nonempty ι] {V : ι → Set X} (hV : Directed (fun x x_1 => x ⊇ x_1) V) (hV_cpct : ∀ (i : ι), IsCompact (V i)) {U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x)
Goal: ∃ i, V i ⊆ U
-- CompactExhaustion.isClosed
theorem CompactExhaustion.isClosed [T2Space X] (K : CompactExhaustion X) (n : ℕ) : IsClosed (K n) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] (K : CompactExhaustion X) (n : ℕ)
Goal: IsClosed (K n)
-- IsCompact.inter
theorem IsCompact.inter [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
    IsCompact (s ∩ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t)
Goal: IsCompact (s ∩ t)
-- image_closure_of_isCompact
theorem image_closure_of_isCompact [T2Space Y] {s : Set X} (hs : IsCompact (closure s)) {f : X → Y}
    (hf : ContinuousOn f (closure s)) : f '' closure s = closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {s : Set X} (hs : IsCompact (closure s)) {f : X → Y} (hf : ContinuousOn f (closure s))
Goal: f '' closure s = closure (f '' s)
-- Continuous.isClosedMap
protected theorem Continuous.isClosedMap [CompactSpace X] [T2Space Y] {f : X → Y}
    (h : Continuous f) : IsClosedMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] [T2Space Y] {f : X → Y} (h : Continuous f)
Goal: IsClosedMap f
-- Continuous.closedEmbedding
theorem Continuous.closedEmbedding [CompactSpace X] [T2Space Y] {f : X → Y} (h : Continuous f)
    (hf : Function.Injective f) : ClosedEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] [T2Space Y] {f : X → Y} (h : Continuous f) (hf : Function.Injective f)
Goal: ClosedEmbedding f
-- QuotientMap.of_surjective_continuous
theorem QuotientMap.of_surjective_continuous [CompactSpace X] [T2Space Y] {f : X → Y}
    (hsurj : Surjective f) (hcont : Continuous f) : QuotientMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] [T2Space Y] {f : X → Y} (hsurj : Function.Surjective f) (hcont : Continuous f)
Goal: QuotientMap f
-- isPreirreducible_iff_subsingleton
theorem isPreirreducible_iff_subsingleton [T2Space X] {S : Set X} :
    IsPreirreducible S ↔ S.Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {S : Set X}
Goal: IsPreirreducible S ↔ S.Subsingleton
-- IsPreirreducible.subsingleton
protected lemma IsPreirreducible.subsingleton [T2Space X] {S : Set X} (h : IsPreirreducible S) :
    S.Subsingleton 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {S : Set X} (h : IsPreirreducible S)
Goal: S.Subsingleton
-- isIrreducible_iff_singleton
theorem isIrreducible_iff_singleton [T2Space X] {S : Set X} : IsIrreducible S ↔ ∃ x, S = {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] {S : Set X}
Goal: IsIrreducible S ↔ ∃ x, S = {x}
-- not_preirreducible_nontrivial_t2
theorem not_preirreducible_nontrivial_t2 (X) [TopologicalSpace X] [PreirreducibleSpace X]
    [Nontrivial X] [T2Space X] : False 
Arguments: (X : Type u_4) [TopologicalSpace X] [PreirreducibleSpace X] [Nontrivial X] [T2Space X]
Goal: False
-- regularSpace_iff
@[mk_iff]
class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
    regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
Arguments: (X : Type u) [TopologicalSpace X]
Goal: RegularSpace X ↔ ∀ {s : Set X} {a : X}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
-- RegularSpace.regular
@[mk_iff]
class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
    regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
Arguments: {X : Type u} [TopologicalSpace X] [self : RegularSpace X] {s : Set X} {a : X}
Goal: IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
-- regularSpace_TFAE
theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
    List.TFAE [RegularSpace X,
      ∀ (s : Set X) x, x ∉ closure s → Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s,
      ∀ (x : X) (s : Set X), s ∈ 𝓝 x → ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x,
      ∀ x : X , (𝓝 x).lift' closure = 𝓝 x] 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
    ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
    ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
-- RegularSpace.of_lift'_closure_le
theorem RegularSpace.of_lift'_closure_le (h : ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x) :
    RegularSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h : ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x)
Goal: RegularSpace X
-- RegularSpace.of_lift'_closure
theorem RegularSpace.of_lift'_closure (h : ∀ x : X, (𝓝 x).lift' closure = 𝓝 x) : RegularSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h : ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x)
Goal: RegularSpace X
-- RegularSpace.of_hasBasis
theorem RegularSpace.of_hasBasis {ι : X → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set X}
    (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
    RegularSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : X → Sort u_3} {p : (a : X) → ι a → Prop} {s : (a : X) → ι a → Set X} (h₁ : ∀ (a : X), (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ (a : X) (i : ι a), p a i → IsClosed (s a i))
Goal: RegularSpace X
-- RegularSpace.of_exists_mem_nhds_isClosed_subset
theorem RegularSpace.of_exists_mem_nhds_isClosed_subset
    (h : ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) : RegularSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h : ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s)
Goal: RegularSpace X
-- disjoint_nhdsSet_nhds
theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {x : X} {s : Set X}
Goal: Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
-- disjoint_nhds_nhdsSet
theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {x : X} {s : Set X}
Goal: Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s
-- exists_mem_nhds_isClosed_subset
theorem exists_mem_nhds_isClosed_subset {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
    ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {x : X} {s : Set X} (h : s ∈ 𝓝 x)
Goal: ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
-- closed_nhds_basis
theorem closed_nhds_basis (x : X) : (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsClosed s) id 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsClosed s) id
-- lift'_nhds_closure
theorem lift'_nhds_closure (x : X) : (𝓝 x).lift' closure = 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] (x : X)
Goal: (𝓝 x).lift' closure = 𝓝 x
-- Filter.HasBasis.nhds_closure
theorem Filter.HasBasis.nhds_closure {ι : Sort*} {x : X} {p : ι → Prop} {s : ι → Set X}
    (h : (𝓝 x).HasBasis p s) : (𝓝 x).HasBasis p fun i => closure (s i) 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {ι : Sort u_3} {x : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s)
Goal: (𝓝 x).HasBasis p fun i => closure (s i)
-- hasBasis_nhds_closure
theorem hasBasis_nhds_closure (x : X) : (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x) closure 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x) closure
-- hasBasis_opens_closure
theorem hasBasis_opens_closure (x : X) : (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsOpen s) closure 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsOpen s) closure
-- IsCompact.exists_isOpen_closure_subset
theorem IsCompact.exists_isOpen_closure_subset {K U : Set X} (hK : IsCompact K) (hU : U ∈ 𝓝ˢ K) :
    ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {K U : Set X} (hK : IsCompact K) (hU : U ∈ 𝓝ˢ K)
Goal: ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U
-- IsCompact.lift'_closure_nhdsSet
theorem IsCompact.lift'_closure_nhdsSet {K : Set X} (hK : IsCompact K) :
    (𝓝ˢ K).lift' closure = 𝓝ˢ K 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {K : Set X} (hK : IsCompact K)
Goal: (𝓝ˢ K).lift' closure = 𝓝ˢ K
-- TopologicalSpace.IsTopologicalBasis.nhds_basis_closure
theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set X)}
    (hB : IsTopologicalBasis B) (x : X) :
    (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ s ∈ B) closure 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {B : Set (Set X)} (hB : TopologicalSpace.IsTopologicalBasis B) (x : X)
Goal: (𝓝 x).HasBasis (fun s => x ∈ s ∧ s ∈ B) closure