ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Equiv.induced_symm
theorem Equiv.induced_symm {α β : Type*} (e : α ≃ β) :
    TopologicalSpace.induced e.symm = TopologicalSpace.coinduced e 
Arguments: {α : Type u_4} {β : Type u_5} (e : α ≃ β)
Goal: TopologicalSpace.induced ⇑e.symm = TopologicalSpace.coinduced ⇑e
-- Equiv.coinduced_symm
theorem Equiv.coinduced_symm {α β : Type*} (e : α ≃ β) :
    TopologicalSpace.coinduced e.symm = TopologicalSpace.induced e 
Arguments: {α : Type u_4} {β : Type u_5} (e : α ≃ β)
Goal: TopologicalSpace.coinduced ⇑e.symm = TopologicalSpace.induced ⇑e
-- continuous_empty_function
theorem continuous_empty_function [TopologicalSpace α] [TopologicalSpace β] [IsEmpty β]
    (f : α → β) : Continuous f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [IsEmpty β] (f : α → β)
Goal: Continuous f
-- le_generateFrom
theorem le_generateFrom {t : TopologicalSpace α} {g : Set (Set α)} (h : ∀ s ∈ g, IsOpen s) :
    t ≤ generateFrom g 
Arguments: {α : Type u} {t : TopologicalSpace α} {g : Set (Set α)} (h : ∀ s ∈ g, IsOpen s)
Goal: t ≤ TopologicalSpace.generateFrom g
-- induced_generateFrom_eq
theorem induced_generateFrom_eq {α β} {b : Set (Set β)} {f : α → β} :
    (generateFrom b).induced f = generateFrom (preimage f '' b) 
Arguments: {α : Type u_1} {β : Type u_2} {b : Set (Set β)} {f : α → β}
Goal: TopologicalSpace.induced f (TopologicalSpace.generateFrom b) = TopologicalSpace.generateFrom (Set.preimage f '' b)
-- le_induced_generateFrom
theorem le_induced_generateFrom {α β} [t : TopologicalSpace α] {b : Set (Set β)} {f : α → β}
    (h : ∀ a : Set β, a ∈ b → IsOpen (f ⁻¹' a)) : t ≤ induced f (generateFrom b) 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] {b : Set (Set β)} {f : α → β} (h : ∀ a ∈ b, IsOpen (f ⁻¹' a))
Goal: t ≤ TopologicalSpace.induced f (TopologicalSpace.generateFrom b)
-- generateFrom_insert_of_generateOpen
lemma generateFrom_insert_of_generateOpen {α : Type*} {s : Set (Set α)} {t : Set α}
    (ht : GenerateOpen s t) : generateFrom (insert t s) = generateFrom s 
Arguments: {α : Type u_1} {s : Set (Set α)} {t : Set α} (ht : TopologicalSpace.GenerateOpen s t)
Goal: TopologicalSpace.generateFrom (insert t s) = TopologicalSpace.generateFrom s
-- generateFrom_insert_univ
@[simp]
lemma generateFrom_insert_univ {α : Type*} {s : Set (Set α)} :
    generateFrom (insert univ s) = generateFrom s 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: TopologicalSpace.generateFrom (insert Set.univ s) = TopologicalSpace.generateFrom s
-- gc_nhds
theorem gc_nhds (a : α) : GaloisConnection (nhdsAdjoint a) fun t => @nhds α t a 
Arguments: {α : Type u} (a : α)
Goal: GaloisConnection (nhdsAdjoint a) fun t => 𝓝 a
-- nhds_mono
theorem nhds_mono {t₁ t₂ : TopologicalSpace α} {a : α} (h : t₁ ≤ t₂) :
    @nhds α t₁ a ≤ @nhds α t₂ a 
Arguments: {α : Type u} {t₁ t₂ : TopologicalSpace α} {a : α} (h : t₁ ≤ t₂)
Goal: 𝓝 a ≤ 𝓝 a
-- le_iff_nhds
theorem le_iff_nhds {α : Type*} (t t' : TopologicalSpace α) :
    t ≤ t' ↔ ∀ x, @nhds α t x ≤ @nhds α t' x 
Arguments: {α : Type u_1} (t t' : TopologicalSpace α)
Goal: t ≤ t' ↔ ∀ (x : α), 𝓝 x ≤ 𝓝 x
-- isOpen_singleton_nhdsAdjoint
theorem isOpen_singleton_nhdsAdjoint {α : Type*} {a b : α} (f : Filter α) (hb : b ≠ a) :
    IsOpen[nhdsAdjoint a f] {b} 
Arguments: {α : Type u_1} {a b : α} (f : Filter α) (hb : b ≠ a)
Goal: IsOpen {b}
-- nhds_nhdsAdjoint_same
theorem nhds_nhdsAdjoint_same (a : α) (f : Filter α) :
    @nhds α (nhdsAdjoint a f) a = pure a ⊔ f 
Arguments: {α : Type u} (a : α) (f : Filter α)
Goal: 𝓝 a = pure a ⊔ f
-- nhds_nhdsAdjoint_of_ne
theorem nhds_nhdsAdjoint_of_ne {a b : α} (f : Filter α) (h : b ≠ a) :
    @nhds α (nhdsAdjoint a f) b = pure b 
Arguments: {α : Type u} {a b : α} (f : Filter α) (h : b ≠ a)
Goal: 𝓝 b = pure b
-- nhdsAdjoint_nhds_of_ne
@[deprecated nhds_nhdsAdjoint_of_ne (since := "2024-02-10")]
theorem nhdsAdjoint_nhds_of_ne (a : α) (f : Filter α) {b : α} (h : b ≠ a) :
    @nhds α (nhdsAdjoint a f) b = pure b 
Arguments: {α : Type u} (a : α) (f : Filter α) {b : α} (h : b ≠ a)
Goal: 𝓝 b = pure b
-- nhds_nhdsAdjoint
theorem nhds_nhdsAdjoint [DecidableEq α] (a : α) (f : Filter α) :
    @nhds α (nhdsAdjoint a f) = update pure a (pure a ⊔ f) 
Arguments: {α : Type u} [DecidableEq α] (a : α) (f : Filter α)
Goal: 𝓝 = Function.update pure a (pure a ⊔ f)
-- le_nhdsAdjoint_iff'
theorem le_nhdsAdjoint_iff' {a : α} {f : Filter α} {t : TopologicalSpace α} :
    t ≤ nhdsAdjoint a f ↔ @nhds α t a ≤ pure a ⊔ f ∧ ∀ b ≠ a, @nhds α t b = pure b 
Arguments: {α : Type u} {a : α} {f : Filter α} {t : TopologicalSpace α}
Goal: t ≤ nhdsAdjoint a f ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b
-- le_nhdsAdjoint_iff
theorem le_nhdsAdjoint_iff {α : Type*} (a : α) (f : Filter α) (t : TopologicalSpace α) :
    t ≤ nhdsAdjoint a f ↔ @nhds α t a ≤ pure a ⊔ f ∧ ∀ b ≠ a, IsOpen[t] {b} 
Arguments: {α : Type u_1} (a : α) (f : Filter α) (t : TopologicalSpace α)
Goal: t ≤ nhdsAdjoint a f ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → IsOpen {b}
-- nhds_iInf
theorem nhds_iInf {ι : Sort*} {t : ι → TopologicalSpace α} {a : α} :
    @nhds α (iInf t) a = ⨅ i, @nhds α (t i) a 
Arguments: {α : Type u} {ι : Sort u_1} {t : ι → TopologicalSpace α} {a : α}
Goal: 𝓝 a = ⨅ i, 𝓝 a
-- nhds_sInf
theorem nhds_sInf {s : Set (TopologicalSpace α)} {a : α} :
    @nhds α (sInf s) a = ⨅ t ∈ s, @nhds α t a 
Arguments: {α : Type u} {s : Set (TopologicalSpace α)} {a : α}
Goal: 𝓝 a = ⨅ t ∈ s, 𝓝 a
-- nhds_inf
theorem nhds_inf {t₁ t₂ : TopologicalSpace α} {a : α} :
    @nhds α (t₁ ⊓ t₂) a = @nhds α t₁ a ⊓ @nhds α t₂ a 
Arguments: {α : Type u} {t₁ t₂ : TopologicalSpace α} {a : α}
Goal: 𝓝 a = 𝓝 a ⊓ 𝓝 a
-- nhds_top
theorem nhds_top {a : α} : @nhds α ⊤ a = ⊤ 
Arguments: {α : Type u} {a : α}
Goal: 𝓝 a = ⊤
-- isOpen_sup
theorem isOpen_sup {t₁ t₂ : TopologicalSpace α} {s : Set α} :
    IsOpen[t₁ ⊔ t₂] s ↔ IsOpen[t₁] s ∧ IsOpen[t₂] s 
Arguments: {α : Type u} {t₁ t₂ : TopologicalSpace α} {s : Set α}
Goal: IsOpen s ↔ IsOpen s ∧ IsOpen s
-- continuous_iff_coinduced_le
theorem continuous_iff_coinduced_le {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β} :
    Continuous[t₁, t₂] f ↔ coinduced f t₁ ≤ t₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β}
Goal: Continuous f ↔ TopologicalSpace.coinduced f t₁ ≤ t₂
-- continuous_iff_le_induced
theorem continuous_iff_le_induced {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β} :
    Continuous[t₁, t₂] f ↔ t₁ ≤ induced f t₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β}
Goal: Continuous f ↔ t₁ ≤ TopologicalSpace.induced f t₂
-- continuous_generateFrom_iff
lemma continuous_generateFrom_iff {t : TopologicalSpace α} {b : Set (Set β)} :
    Continuous[t, generateFrom b] f ↔ ∀ s ∈ b, IsOpen (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α} {b : Set (Set β)}
Goal: Continuous f ↔ ∀ s ∈ b, IsOpen (f ⁻¹' s)
-- continuous_induced_dom
@[continuity, fun_prop]
theorem continuous_induced_dom {t : TopologicalSpace β} : Continuous[induced f t, t] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace β}
Goal: Continuous f
-- continuous_induced_rng
theorem continuous_induced_rng {g : γ → α} {t₂ : TopologicalSpace β} {t₁ : TopologicalSpace γ} :
    Continuous[t₁, induced f t₂] g ↔ Continuous[t₁, t₂] (f ∘ g) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} {f : α → β} {g : γ → α} {t₂ : TopologicalSpace β} {t₁ : TopologicalSpace γ}
Goal: Continuous g ↔ Continuous (f ∘ g)
-- continuous_coinduced_rng
theorem continuous_coinduced_rng {t : TopologicalSpace α} :
    Continuous[t, coinduced f t] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α}
Goal: Continuous f
-- continuous_coinduced_dom
theorem continuous_coinduced_dom {g : β → γ} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace γ} :
    Continuous[coinduced f t₁, t₂] g ↔ Continuous[t₁, t₂] (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} {f : α → β} {g : β → γ} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace γ}
Goal: Continuous g ↔ Continuous (g ∘ f)
-- continuous_le_dom
theorem continuous_le_dom {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₁)
    (h₂ : Continuous[t₁, t₃] f) : Continuous[t₂, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₁) (h₂ : Continuous f)
Goal: Continuous f
-- continuous_le_rng
theorem continuous_le_rng {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₃)
    (h₂ : Continuous[t₁, t₂] f) : Continuous[t₁, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₃) (h₂ : Continuous f)
Goal: Continuous f
-- continuous_sup_dom
theorem continuous_sup_dom {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} :
    Continuous[t₁ ⊔ t₂, t₃] f ↔ Continuous[t₁, t₃] f ∧ Continuous[t₂, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
Goal: Continuous f ↔ Continuous f ∧ Continuous f
-- continuous_sup_rng_left
theorem continuous_sup_rng_left {t₁ : TopologicalSpace α} {t₃ t₂ : TopologicalSpace β} :
    Continuous[t₁, t₂] f → Continuous[t₁, t₂ ⊔ t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₃ t₂ : TopologicalSpace β}
Goal: Continuous f → Continuous f
-- continuous_sup_rng_right
theorem continuous_sup_rng_right {t₁ : TopologicalSpace α} {t₃ t₂ : TopologicalSpace β} :
    Continuous[t₁, t₃] f → Continuous[t₁, t₂ ⊔ t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₃ t₂ : TopologicalSpace β}
Goal: Continuous f → Continuous f
-- continuous_sSup_dom
theorem continuous_sSup_dom {T : Set (TopologicalSpace α)} {t₂ : TopologicalSpace β} :
    Continuous[sSup T, t₂] f ↔ ∀ t ∈ T, Continuous[t, t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {T : Set (TopologicalSpace α)} {t₂ : TopologicalSpace β}
Goal: Continuous f ↔ ∀ t ∈ T, Continuous f
-- continuous_sSup_rng
theorem continuous_sSup_rng {t₁ : TopologicalSpace α} {t₂ : Set (TopologicalSpace β)}
    {t : TopologicalSpace β} (h₁ : t ∈ t₂) (hf : Continuous[t₁, t] f) :
    Continuous[t₁, sSup t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ : Set (TopologicalSpace β)} {t : TopologicalSpace β} (h₁ : t ∈ t₂) (hf : Continuous f)
Goal: Continuous f
-- continuous_iSup_dom
theorem continuous_iSup_dom {t₁ : ι → TopologicalSpace α} {t₂ : TopologicalSpace β} :
    Continuous[iSup t₁, t₂] f ↔ ∀ i, Continuous[t₁ i, t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {ι : Sort u_2} {t₁ : ι → TopologicalSpace α} {t₂ : TopologicalSpace β}
Goal: Continuous f ↔ ∀ (i : ι), Continuous f
-- continuous_iSup_rng
theorem continuous_iSup_rng {t₁ : TopologicalSpace α} {t₂ : ι → TopologicalSpace β} {i : ι}
    (h : Continuous[t₁, t₂ i] f) : Continuous[t₁, iSup t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {ι : Sort u_2} {t₁ : TopologicalSpace α} {t₂ : ι → TopologicalSpace β} {i : ι} (h : Continuous f)
Goal: Continuous f
-- continuous_inf_rng
theorem continuous_inf_rng {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β} :
    Continuous[t₁, t₂ ⊓ t₃] f ↔ Continuous[t₁, t₂] f ∧ Continuous[t₁, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
Goal: Continuous f ↔ Continuous f ∧ Continuous f
-- continuous_inf_dom_left
theorem continuous_inf_dom_left {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} :
    Continuous[t₁, t₃] f → Continuous[t₁ ⊓ t₂, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
Goal: Continuous f → Continuous f
-- continuous_inf_dom_right
theorem continuous_inf_dom_right {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} :
    Continuous[t₂, t₃] f → Continuous[t₁ ⊓ t₂, t₃] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
Goal: Continuous f → Continuous f
-- continuous_sInf_dom
theorem continuous_sInf_dom {t₁ : Set (TopologicalSpace α)} {t₂ : TopologicalSpace β}
    {t : TopologicalSpace α} (h₁ : t ∈ t₁) :
    Continuous[t, t₂] f → Continuous[sInf t₁, t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : Set (TopologicalSpace α)} {t₂ : TopologicalSpace β} {t : TopologicalSpace α} (h₁ : t ∈ t₁)
Goal: Continuous f → Continuous f
-- continuous_sInf_rng
theorem continuous_sInf_rng {t₁ : TopologicalSpace α} {T : Set (TopologicalSpace β)} :
    Continuous[t₁, sInf T] f ↔ ∀ t ∈ T, Continuous[t₁, t] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {T : Set (TopologicalSpace β)}
Goal: Continuous f ↔ ∀ t ∈ T, Continuous f
-- continuous_iInf_dom
theorem continuous_iInf_dom {t₁ : ι → TopologicalSpace α} {t₂ : TopologicalSpace β} {i : ι} :
    Continuous[t₁ i, t₂] f → Continuous[iInf t₁, t₂] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {ι : Sort u_2} {t₁ : ι → TopologicalSpace α} {t₂ : TopologicalSpace β} {i : ι}
Goal: Continuous f → Continuous f
-- continuous_iInf_rng
theorem continuous_iInf_rng {t₁ : TopologicalSpace α} {t₂ : ι → TopologicalSpace β} :
    Continuous[t₁, iInf t₂] f ↔ ∀ i, Continuous[t₁, t₂ i] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {ι : Sort u_2} {t₁ : TopologicalSpace α} {t₂ : ι → TopologicalSpace β}
Goal: Continuous f ↔ ∀ (i : ι), Continuous f
-- continuous_bot
@[continuity, fun_prop]
theorem continuous_bot {t : TopologicalSpace β} : Continuous[⊥, t] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace β}
Goal: Continuous f
-- continuous_top
@[continuity, fun_prop]
theorem continuous_top {t : TopologicalSpace α} : Continuous[t, ⊤] f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α}
Goal: Continuous f
-- continuous_id_iff_le
theorem continuous_id_iff_le {t t' : TopologicalSpace α} : Continuous[t, t'] id ↔ t ≤ t' 
Arguments: {α : Type u} {t t' : TopologicalSpace α}
Goal: Continuous id ↔ t ≤ t'
-- continuous_id_of_le
theorem continuous_id_of_le {t t' : TopologicalSpace α} (h : t ≤ t') : Continuous[t, t'] id 
Arguments: {α : Type u} {t t' : TopologicalSpace α} (h : t ≤ t')
Goal: Continuous id
-- mem_nhds_induced
theorem mem_nhds_induced [T : TopologicalSpace α] (f : β → α) (a : β) (s : Set β) :
    s ∈ @nhds β (TopologicalSpace.induced f T) a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s 
Arguments: {α : Type u} {β : Type v} [T : TopologicalSpace α] (f : β → α) (a : β) (s : Set β)
Goal: s ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s
-- nhds_induced
theorem nhds_induced [T : TopologicalSpace α] (f : β → α) (a : β) :
    @nhds β (TopologicalSpace.induced f T) a = comap f (𝓝 (f a)) 
Arguments: {α : Type u} {β : Type v} [T : TopologicalSpace α] (f : β → α) (a : β)
Goal: 𝓝 a = Filter.comap f (𝓝 (f a))
-- induced_iff_nhds_eq
theorem induced_iff_nhds_eq [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : β → α) :
    tβ = tα.induced f ↔ ∀ b, 𝓝 b = comap f (𝓝 <| f b) 
Arguments: {α : Type u} {β : Type v} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : β → α)
Goal: tβ = TopologicalSpace.induced f tα ↔ ∀ (b : β), 𝓝 b = Filter.comap f (𝓝 (f b))
-- map_nhds_induced_of_surjective
theorem map_nhds_induced_of_surjective [T : TopologicalSpace α] {f : β → α} (hf : Surjective f)
    (a : β) : map f (@nhds β (TopologicalSpace.induced f T) a) = 𝓝 (f a) 
Arguments: {α : Type u} {β : Type v} [T : TopologicalSpace α] {f : β → α} (hf : Function.Surjective f) (a : β)
Goal: Filter.map f (𝓝 a) = 𝓝 (f a)
-- isOpen_induced_eq
theorem isOpen_induced_eq {s : Set α} :
    IsOpen[induced f t] s ↔ s ∈ preimage f '' { s | IsOpen s } 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set α}
Goal: IsOpen s ↔ s ∈ Set.preimage f '' {s | IsOpen s}
-- isOpen_induced
theorem isOpen_induced {s : Set β} (h : IsOpen s) : IsOpen[induced f t] (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set β} (h : IsOpen s)
Goal: IsOpen (f ⁻¹' s)
-- map_nhds_induced_eq
theorem map_nhds_induced_eq (a : α) : map f (@nhds α (induced f t) a) = 𝓝[range f] f a 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} (a : α)
Goal: Filter.map f (𝓝 a) = 𝓝[Set.range f] f a
-- map_nhds_induced_of_mem
theorem map_nhds_induced_of_mem {a : α} (h : range f ∈ 𝓝 (f a)) :
    map f (@nhds α (induced f t) a) = 𝓝 (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {a : α} (h : Set.range f ∈ 𝓝 (f a))
Goal: Filter.map f (𝓝 a) = 𝓝 (f a)
-- closure_induced
theorem closure_induced {f : α → β} {a : α} {s : Set α} :
    a ∈ @closure α (t.induced f) s ↔ f a ∈ closure (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {a : α} {s : Set α}
Goal: a ∈ closure s ↔ f a ∈ closure (f '' s)
-- isClosed_induced_iff'
theorem isClosed_induced_iff' {f : α → β} {s : Set α} :
    IsClosed[t.induced f] s ↔ ∀ a, f a ∈ closure (f '' s) → a ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set α}
Goal: IsClosed s ↔ ∀ (a : α), f a ∈ closure (f '' s) → a ∈ s
-- isOpen_singleton_true
@[simp]
theorem isOpen_singleton_true : IsOpen ({True} : Set Prop) 
Goal: IsOpen {True}
-- nhds_true
@[simp]
theorem nhds_true : 𝓝 True = pure True 
Goal: 𝓝 True = pure True
-- nhds_false
@[simp]
theorem nhds_false : 𝓝 False = ⊤ 
Goal: 𝓝 False = ⊤
-- tendsto_nhds_true
theorem tendsto_nhds_true {l : Filter α} {p : α → Prop} :
    Tendsto p l (𝓝 True) ↔ ∀ᶠ x in l, p x 
Arguments: {α : Type u_1} {l : Filter α} {p : α → Prop}
Goal: Filter.Tendsto p l (𝓝 True) ↔ ∀ᶠ (x : α) in l, p x
-- tendsto_nhds_Prop
theorem tendsto_nhds_Prop {l : Filter α} {p : α → Prop} {q : Prop} :
    Tendsto p l (𝓝 q) ↔ (q → ∀ᶠ x in l, p x) 
Arguments: {α : Type u_1} {l : Filter α} {p : α → Prop} {q : Prop}
Goal: Filter.Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x
-- continuous_Prop
theorem continuous_Prop {p : α → Prop} : Continuous p ↔ IsOpen { x | p x } 
Arguments: {α : Type u_1} [TopologicalSpace α] {p : α → Prop}
Goal: Continuous p ↔ IsOpen {x | p x}
-- isOpen_iff_continuous_mem
theorem isOpen_iff_continuous_mem {s : Set α} : IsOpen s ↔ Continuous (· ∈ s) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α}
Goal: IsOpen s ↔ Continuous fun x => x ∈ s
-- generateFrom_union
theorem generateFrom_union (a₁ a₂ : Set (Set α)) :
    generateFrom (a₁ ∪ a₂) = generateFrom a₁ ⊓ generateFrom a₂ 
Arguments: {α : Type u} (a₁ a₂ : Set (Set α))
Goal: TopologicalSpace.generateFrom (a₁ ∪ a₂) = TopologicalSpace.generateFrom a₁ ⊓ TopologicalSpace.generateFrom a₂
-- setOf_isOpen_sup
theorem setOf_isOpen_sup (t₁ t₂ : TopologicalSpace α) :
    { s | IsOpen[t₁ ⊔ t₂] s } = { s | IsOpen[t₁] s } ∩ { s | IsOpen[t₂] s } 
Arguments: {α : Type u} (t₁ t₂ : TopologicalSpace α)
Goal: {s | IsOpen s} = {s | IsOpen s} ∩ {s | IsOpen s}
-- generateFrom_iUnion
theorem generateFrom_iUnion {f : ι → Set (Set α)} :
    generateFrom (⋃ i, f i) = ⨅ i, generateFrom (f i) 
Arguments: {α : Type u} {ι : Sort v} {f : ι → Set (Set α)}
Goal: TopologicalSpace.generateFrom (⋃ i, f i) = ⨅ i, TopologicalSpace.generateFrom (f i)
-- setOf_isOpen_iSup
theorem setOf_isOpen_iSup {t : ι → TopologicalSpace α} :
    { s | IsOpen[⨆ i, t i] s } = ⋂ i, { s | IsOpen[t i] s } 
Arguments: {α : Type u} {ι : Sort v} {t : ι → TopologicalSpace α}
Goal: {s | IsOpen s} = ⋂ i, {s | IsOpen s}
-- generateFrom_sUnion
theorem generateFrom_sUnion {S : Set (Set (Set α))} :
    generateFrom (⋃₀ S) = ⨅ s ∈ S, generateFrom s 
Arguments: {α : Type u} {S : Set (Set (Set α))}
Goal: TopologicalSpace.generateFrom (⋃₀ S) = ⨅ s ∈ S, TopologicalSpace.generateFrom s
-- setOf_isOpen_sSup
theorem setOf_isOpen_sSup {T : Set (TopologicalSpace α)} :
    { s | IsOpen[sSup T] s } = ⋂ t ∈ T, { s | IsOpen[t] s } 
Arguments: {α : Type u} {T : Set (TopologicalSpace α)}
Goal: {s | IsOpen s} = ⋂ t ∈ T, {s | IsOpen s}
-- generateFrom_union_isOpen
theorem generateFrom_union_isOpen (a b : TopologicalSpace α) :
    generateFrom ({ s | IsOpen[a] s } ∪ { s | IsOpen[b] s }) = a ⊓ b 
Arguments: {α : Type u} (a b : TopologicalSpace α)
Goal: TopologicalSpace.generateFrom ({s | IsOpen s} ∪ {s | IsOpen s}) = a ⊓ b
-- generateFrom_iUnion_isOpen
theorem generateFrom_iUnion_isOpen (f : ι → TopologicalSpace α) :
    generateFrom (⋃ i, { s | IsOpen[f i] s }) = ⨅ i, f i 
Arguments: {α : Type u} {ι : Sort v} (f : ι → TopologicalSpace α)
Goal: TopologicalSpace.generateFrom (⋃ i, {s | IsOpen s}) = ⨅ i, f i
-- generateFrom_inter
theorem generateFrom_inter (a b : TopologicalSpace α) :
    generateFrom ({ s | IsOpen[a] s } ∩ { s | IsOpen[b] s }) = a ⊔ b 
Arguments: {α : Type u} (a b : TopologicalSpace α)
Goal: TopologicalSpace.generateFrom ({s | IsOpen s} ∩ {s | IsOpen s}) = a ⊔ b
-- generateFrom_iInter
theorem generateFrom_iInter (f : ι → TopologicalSpace α) :
    generateFrom (⋂ i, { s | IsOpen[f i] s }) = ⨆ i, f i 
Arguments: {α : Type u} {ι : Sort v} (f : ι → TopologicalSpace α)
Goal: TopologicalSpace.generateFrom (⋂ i, {s | IsOpen s}) = ⨆ i, f i
-- generateFrom_iInter_of_generateFrom_eq_self
theorem generateFrom_iInter_of_generateFrom_eq_self (f : ι → Set (Set α))
    (hf : ∀ i, { s | IsOpen[generateFrom (f i)] s } = f i) :
    generateFrom (⋂ i, f i) = ⨆ i, generateFrom (f i) 
Arguments: {α : Type u} {ι : Sort v} (f : ι → Set (Set α)) (hf : ∀ (i : ι), {s | IsOpen s} = f i)
Goal: TopologicalSpace.generateFrom (⋂ i, f i) = ⨆ i, TopologicalSpace.generateFrom (f i)
-- isOpen_iSup_iff
theorem isOpen_iSup_iff {s : Set α} : IsOpen[⨆ i, t i] s ↔ ∀ i, IsOpen[t i] s 
Arguments: {α : Type u} {ι : Sort v} {t : ι → TopologicalSpace α} {s : Set α}
Goal: IsOpen s ↔ ∀ (i : ι), IsOpen s
-- isClosed_iSup_iff
theorem isClosed_iSup_iff {s : Set α} : IsClosed[⨆ i, t i] s ↔ ∀ i, IsClosed[t i] s 
Arguments: {α : Type u} {ι : Sort v} {t : ι → TopologicalSpace α} {s : Set α}
Goal: IsClosed s ↔ ∀ (i : ι), IsClosed s
-- Path.Homotopy.coeFn_injective
theorem coeFn_injective : @Function.Injective (Homotopy p₀ p₁) (I × I → X) (⇑) 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁}
Goal: Function.Injective DFunLike.coe
-- Path.Homotopy.source
@[simp]
theorem source (F : Homotopy p₀ p₁) (t : I) : F (t, 0) = x₀ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑I)
Goal: F (t, 0) = x₀
-- Path.Homotopy.target
@[simp]
theorem target (F : Homotopy p₀ p₁) (t : I) : F (t, 1) = x₁ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑I)
Goal: F (t, 1) = x₁
-- Path.Homotopy.eval_zero
@[simp]
theorem eval_zero (F : Homotopy p₀ p₁) : F.eval 0 = p₀ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁)
Goal: F.eval 0 = p₀
-- Path.Homotopy.eval_one
@[simp]
theorem eval_one (F : Homotopy p₀ p₁) : F.eval 1 = p₁ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁)
Goal: F.eval 1 = p₁
-- Path.Homotopy.refl_toFun
@[simps!]
def refl (p : Path x₀ x₁) : Homotopy p p 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (x : ↑I × ↑I)
Goal: (Path.Homotopy.refl p) x = p x.2
-- Path.Homotopy.refl_apply
@[simps!]
def refl (p : Path x₀ x₁) : Homotopy p p 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (x : ↑I × ↑I)
Goal: (Path.Homotopy.refl p) x = p x.2
-- Path.Homotopy.symm_toFun
@[simps!]
def symm (F : Homotopy p₀ p₁) : Homotopy p₁ p₀ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (x : ↑I × ↑I)
Goal: F.symm x = F (σ x.1, x.2)
-- Path.Homotopy.symm_apply
@[simps!]
def symm (F : Homotopy p₀ p₁) : Homotopy p₁ p₀ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (x : ↑I × ↑I)
Goal: F.symm x = F (σ x.1, x.2)
-- Path.Homotopy.symm_symm
@[simp]
theorem symm_symm (F : Homotopy p₀ p₁) : F.symm.symm = F 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁)
Goal: F.symm.symm = F
-- Path.Homotopy.symm_bijective
theorem symm_bijective : Function.Bijective (Homotopy.symm : Homotopy p₀ p₁ → Homotopy p₁ p₀) 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁}
Goal: Function.Bijective Path.Homotopy.symm