Loogle!

Result

Found 523 declarations mentioning Filter.principal. Of these, only the first 200 are shown.

• Filter.principal 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} (s : Set α) : Filter α
• Filter.hnot_def 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {f : Filter α} : ￢f = Filter.principal f.kerᶜ
• Filter.mem_principal 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {s t : Set α} : s ∈ Filter.principal t ↔ t ⊆ s
• Set.Nonempty.principal_neBot 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} : s.Nonempty → (Filter.principal s).NeBot
• Filter.principal_neBot_iff 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} : (Filter.principal s).NeBot ↔ s.Nonempty
• Filter.principal_univ 📋 Mathlib.Order.Filter.Basic
  {α : Type u} : Filter.principal Set.univ = ⊤
• Filter.mem_principal_self 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (s : Set α) : s ∈ Filter.principal s
• Filter.principal_empty 📋 Mathlib.Order.Filter.Basic
  {α : Type u} : Filter.principal ∅ = ⊥
• Filter.eventually_mem_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (s : Set α) : ∀ᶠ (x : α) in Filter.principal s, x ∈ s
• Filter.join_principal_eq_sSup 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set (Filter α)} : (Filter.principal s).join = sSup s
• Filter.principal_eq_iff_eq 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} : Filter.principal s = Filter.principal t ↔ s = t
• Filter.generate_singleton 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (s : Set α) : Filter.generate {s} = Filter.principal s
• Filter.hnot_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} : ￢Filter.principal s = Filter.principal sᶜ
• Set.EqOn.eventuallyEq 📋 Mathlib.Order.Filter.Basic
  {α : Type u_1} {β : Type u_2} {s : Set α} {f g : α → β} (h : Set.EqOn f g s) : f =ᶠ[Filter.principal s] g
• Filter.eventuallyEq_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {s : Set α} {f g : α → β} : f =ᶠ[Filter.principal s] g ↔ Set.EqOn f g s
• Filter.isCompl_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (s : Set α) : IsCompl (Filter.principal s) (Filter.principal sᶜ)
• Filter.principal_eq_bot_iff 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} : Filter.principal s = ⊥ ↔ s = ∅
• Filter.eventually_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {a : Set α} {p : α → Prop} : (∀ᶠ (x : α) in Filter.principal a, p x) ↔ ∀ x ∈ a, p x
• Filter.frequently_iff_neBot 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {l : Filter α} {p : α → Prop} : (∃ᶠ (x : α) in l, p x) ↔ (l ⊓ Filter.principal {x | p x}).NeBot
• Filter.iSup_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {ι : Sort w} {s : ι → Set α} : ⨆ x, Filter.principal (s x) = Filter.principal (⋃ i, s i)
• Filter.frequently_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {a : Set α} {p : α → Prop} : (∃ᶠ (x : α) in Filter.principal a, p x) ↔ ∃ x ∈ a, p x
• Filter.inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} : Filter.principal s ⊓ Filter.principal t = Filter.principal (s ∩ t)
• Filter.principal_sdiff_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} : Filter.principal s \ Filter.principal t = Filter.principal (s \ t)
• Filter.sup_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} : Filter.principal s ⊔ Filter.principal t = Filter.principal (s ∪ t)
• Filter.frequently_mem_iff_neBot 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {l : Filter α} {s : Set α} : (∃ᶠ (x : α) in l, x ∈ s) ↔ (l ⊓ Filter.principal s).NeBot
• Filter.eventuallySubset_iff_mem_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ t ∈ l ⊓ Filter.principal s
• Filter.le_principal_iff 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} {f : Filter α} : f ≤ Filter.principal s ↔ s ∈ f
• Filter.monotone_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} : Monotone Filter.principal
• Filter.principal_mono 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} : Filter.principal s ≤ Filter.principal t ↔ s ⊆ t
• Filter.set_eventuallyLE_iff_mem_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ t ∈ l ⊓ Filter.principal s
• Filter.Iic_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (s : Set α) : Set.Iic (Filter.principal s) = {l | s ∈ l}
• Filter.eventuallyEqSet_iff_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s =ᶠ[l] t ↔ l ⊓ Filter.principal s = l ⊓ Filter.principal t
• Filter.set_eventuallyEq_iff_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s =ᶠ[l] t ↔ l ⊓ Filter.principal s = l ⊓ Filter.principal t
• Filter.eventually_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {p : α → Prop} {s : Set α} : (∀ᶠ (x : α) in f ⊓ Filter.principal s, p x) ↔ ∀ᶠ (x : α) in f, x ∈ s → p x
• Filter.mem_of_eq_bot 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} (h : f ⊓ Filter.principal sᶜ = ⊥) : s ∈ f
• Filter.Frequently.inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f, x ∈ s ∧ p x) → ∃ᶠ (x : α) in f ⊓ Filter.principal s, p x
• Filter.Frequently.of_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f ⊓ Filter.principal s, p x) → ∃ᶠ (x : α) in f, x ∈ s ∧ p x
• Filter.frequently_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f ⊓ Filter.principal s, p x) ↔ ∃ᶠ (x : α) in f, x ∈ s ∧ p x
• Filter.inf_principal_eq_bot 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} : f ⊓ Filter.principal s = ⊥ ↔ sᶜ ∈ f
• Filter.principal_le_iff 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s : Set α} {f : Filter α} : Filter.principal s ≤ f ↔ ∀ V ∈ f, s ⊆ V
• Filter.Eventually.forall_mem 📋 Mathlib.Order.Filter.Basic
  {α : Type u_2} {f : Filter α} {s : Set α} {P : α → Prop} (hP : ∀ᶠ (x : α) in f, P x) (hf : Filter.principal s ≤ f) (x : α) : x ∈ s → P x
• Filter.eventuallySubset_iff_inf_principal_le 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ l ⊓ Filter.principal s ≤ l ⊓ Filter.principal t
• Filter.set_eventuallyLE_iff_inf_principal_le 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {s t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ l ⊓ Filter.principal s ≤ l ⊓ Filter.principal t
• Filter.eventuallyEq_inf_principal_iff 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {F : Filter α} {s : Set α} {f g : α → β} : f =ᶠ[F ⊓ Filter.principal s] g ↔ ∀ᶠ (x : α) in F, x ∈ s → f x = g x
• Filter.diff_mem_inf_principal_compl 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} (hs : s ∈ f) (t : Set α) : s \ t ∈ f ⊓ Filter.principal tᶜ
• Filter.iSup_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {ι : Sort x} (f : ι → Filter α) (s : Set α) : ⨆ i, f i ⊓ Filter.principal s = (⨆ i, f i) ⊓ Filter.principal s
• Filter.sdiff_mem_inf_principal_compl 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s : Set α} (hs : s ∈ f) (t : Set α) : s \ t ∈ f ⊓ Filter.principal tᶜ
• Filter.mem_inf_principal' 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s t : Set α} : s ∈ f ⊓ Filter.principal t ↔ tᶜ ∪ s ∈ f
• Filter.generate_eq_biInf 📋 Mathlib.Order.Filter.Basic
  {α : Type u} (S : Set (Set α)) : Filter.generate S = ⨅ s ∈ S, Filter.principal s
• Filter.mem_inf_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {f : Filter α} {s t : Set α} : s ∈ f ⊓ Filter.principal t ↔ {x | x ∈ t → x ∈ s} ∈ f
• Filter.principal_singleton 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (a : α) : Filter.principal {a} = pure a
• Filter.map_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) : Filter.map f ⊤ = Filter.principal (Set.range f)
• Filter.comap_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} {t : Set β} : Filter.comap m (Filter.principal t) = Filter.principal (m ⁻¹' t)
• Filter.kernMap_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} {s : Set α} : Filter.kernMap m (Filter.principal s) = Filter.principal (Set.kernImage m s)
• Filter.map_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} : Filter.map f (Filter.principal s) = Filter.principal (f '' s)
• Filter.comap_inf_principal_range 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} : Filter.comap m (g ⊓ Filter.principal (Set.range m)) = Filter.comap m g
• Filter.comap_pure 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} {b : β} : Filter.comap m (pure b) = Filter.principal (m ⁻¹' {b})
• Filter.map_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : Filter β) (m : α → β) : Filter.map m (Filter.comap m f) = f ⊓ Filter.principal (Set.range m)
• Set.InjOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn m s → Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s))
• Filter.filter_injOn_Iic_iff_injOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s)) ↔ Set.InjOn m s
• Filter.pure_le_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {s : Set α} (a : α) : pure a ≤ Filter.principal s ↔ a ∈ s
• Set.LeftInvOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {g : β → α} (hfg : Set.LeftInvOn g f s) : Set.LeftInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal s))
• Set.RightInvOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {t : Set β} {f : α → β} {g : β → α} (hfg : Set.RightInvOn g f t) : Set.RightInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal t))
• Filter.comap_inf_principal_neBot_of_image_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hf : f.NeBot) {s : Set α} (hs : m '' s ∈ f) : (Filter.comap m f ⊓ Filter.principal s).NeBot
• Filter.bind_inf_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : α → Filter β} {s : Set β} : (f.bind fun x => g x ⊓ Filter.principal s) = f.bind g ⊓ Filter.principal s
• Filter.biSup_pure_eq_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (s : Set α) : ⨆ a ∈ s, pure a = Filter.principal s
• Set.SurjOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn m s t → Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t))
• Filter.map_surjOn_Iic_iff_surjOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t)) ↔ Set.SurjOn m s t
• Filter.principal_bind 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → Filter β} : (Filter.principal s).bind f = ⨆ x ∈ s, f x
• Filter.principal_eq_map_coe_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (s : Set α) : Filter.principal s = Filter.map Subtype.val ⊤
• Filter.comap_coe_neBot_of_le_principal 📋 Mathlib.Order.Filter.Map
  {γ : Type u_3} {s : Set γ} {l : Filter γ} [h : l.NeBot] (h' : l ≤ Filter.principal s) : (Filter.comap Subtype.val l).NeBot
• Filter.principal_subtype 📋 Mathlib.Order.Filter.Map
  {α : Type u_6} (s : Set α) (t : Set ↑s) : Filter.principal t = Filter.comap Subtype.val (Filter.principal (Subtype.val '' t))
• Filter.map_comap_setCoe_val 📋 Mathlib.Order.Filter.Map
  {β : Type u_2} (f : Filter β) (s : Set β) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s
• Filter.subtype_coe_map_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (s : Set α) (f : Filter α) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s
• Filter.inf_principal_eq_bot_iff_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {F : Filter α} {s : Set α} : F ⊓ Filter.principal s = ⊥ ↔ Filter.comap Subtype.val F = ⊥
• Filter.hasBasis_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} (t : Set α) : (Filter.principal t).HasBasis (fun x => True) fun x => t
• Filter.HasBasis.eq_iInf 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {s : ι → Set α} (h : l.HasBasis (fun x => True) s) : l = ⨅ i, Filter.principal (s i)
• Filter.notMem_iff_inf_principal_compl 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {f : Filter α} {s : Set α} : s ∉ f ↔ (f ⊓ Filter.principal sᶜ).NeBot
• Filter.inf_principal_neBot_iff 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {l : Filter α} {s : Set α} : (l ⊓ Filter.principal s).NeBot ↔ ∀ U ∈ l, (U ∩ s).Nonempty
• Filter.mem_iff_inf_principal_compl 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {f : Filter α} {s : Set α} : s ∈ f ↔ f ⊓ Filter.principal sᶜ = ⊥
• Filter.HasBasis.eq_biInf 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l = ⨅ i, ⨅ (_ : p i), Filter.principal (s i)
• Filter.HasBasis.inf_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (s' : Set α) : (l ⊓ Filter.principal s').HasBasis p fun i => s i ∩ s'
• Filter.HasBasis.principal_inf 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (s' : Set α) : (Filter.principal s' ⊓ l).HasBasis p fun i => s' ∩ s i
• Filter.HasBasis.sup_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (t : Set α) : (l ⊔ Filter.principal t).HasBasis p fun i => s i ∪ t
• Filter.hasBasis_iInf_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {s : ι → Set α} (h : Directed (fun x1 x2 => x1 ⊇ x2) s) [Nonempty ι] : (⨅ i, Filter.principal (s i)).HasBasis (fun x => True) s
• Filter.HasBasis.inf_principal_neBot_iff 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {t : Set α} : (l ⊓ Filter.principal t).NeBot ↔ ∀ ⦃i : ι⦄, p i → (s i ∩ t).Nonempty
• Filter.disjoint_principal_left 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {f : Filter α} {s : Set α} : Disjoint (Filter.principal s) f ↔ sᶜ ∈ f
• Filter.disjoint_principal_right 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {f : Filter α} {s : Set α} : Disjoint f (Filter.principal s) ↔ sᶜ ∈ f
• FilterBasis.eq_iInf_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} (B : FilterBasis α) : B.filter = ⨅ s, Filter.principal ↑s
• Filter.le_iff_forall_inf_principal_compl 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {f g : Filter α} : f ≤ g ↔ ∀ V ∈ g, f ⊓ Filter.principal Vᶜ = ⊥
• Filter.HasAntitoneBasis.iInf_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Type u_6} [Preorder ι] [Nonempty ι] [IsDirectedOrder ι] {s : ι → Set α} (hs : Antitone s) : (⨅ i, Filter.principal (s i)).HasAntitoneBasis s
• Filter.HasBasis.principal_prod 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι' : Sort u_4} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} (sa : Set α) (h : lb.HasBasis pb sb) : (Filter.principal sa ×ˢ lb).HasBasis pb fun x => sa ×ˢ sb x
• Filter.HasBasis.prod_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort u_3} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} (h : la.HasBasis pa sa) (sb : Set β) : (la ×ˢ Filter.principal sb).HasBasis pa fun x => sa x ×ˢ sb
• Disjoint.filter_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {s t : Set α} : Disjoint s t → Disjoint (Filter.principal s) (Filter.principal t)
• Filter.hasBasis_biInf_principal' 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {ι : Type u_5} {p : ι → Prop} {s : ι → Set α} (h : ∀ (i : ι), p i → ∀ (j : ι), p j → ∃ k, p k ∧ s k ⊆ s i ∧ s k ⊆ s j) (ne : ∃ i, p i) : (⨅ i, ⨅ (_ : p i), Filter.principal (s i)).HasBasis p s
• Filter.disjoint_principal_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {s t : Set α} : Disjoint (Filter.principal s) (Filter.principal t) ↔ Disjoint s t
• Filter.hasBasis_biInf_principal 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {s : β → Set α} {S : Set β} (h : DirectedOn (s ⁻¹'o fun x1 x2 => x1 ⊇ x2) S) (ne : S.Nonempty) : (⨅ i ∈ S, Filter.principal (s i)).HasBasis (fun i => i ∈ S) s
• IsBot.atBot_eq 📋 Mathlib.Order.Filter.AtTopBot.Defs
  {α : Type u_2} [Preorder α] {a : α} (ha : IsBot a) : Filter.atBot = Filter.principal (Set.Iic a)
• IsTop.atTop_eq 📋 Mathlib.Order.Filter.AtTopBot.Defs
  {α : Type u_2} [Preorder α] {a : α} (ha : IsTop a) : Filter.atTop = Filter.principal (Set.Ici a)
• Filter.disjoint_atBot_principal_Ioi 📋 Mathlib.Order.Filter.AtTopBot.Disjoint
  {α : Type u_1} [Preorder α] (x : α) : Disjoint Filter.atBot (Filter.principal (Set.Ioi x))
• Filter.disjoint_atTop_principal_Iio 📋 Mathlib.Order.Filter.AtTopBot.Disjoint
  {α : Type u_1} [Preorder α] (x : α) : Disjoint Filter.atTop (Filter.principal (Set.Iio x))
• Filter.disjoint_atBot_principal_Ici 📋 Mathlib.Order.Filter.AtTopBot.Disjoint
  {α : Type u_1} [Preorder α] [NoBotOrder α] (x : α) : Disjoint Filter.atBot (Filter.principal (Set.Ici x))
• Filter.disjoint_atTop_principal_Iic 📋 Mathlib.Order.Filter.AtTopBot.Disjoint
  {α : Type u_1} [Preorder α] [NoTopOrder α] (x : α) : Disjoint Filter.atTop (Filter.principal (Set.Iic x))
• Set.MapsTo.tendsto 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f : α → β} (h : Set.MapsTo f s t) : Filter.Tendsto f (Filter.principal s) (Filter.principal t)
• Filter.tendsto_principal 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter α} {s : Set β} : Filter.Tendsto f l (Filter.principal s) ↔ ∀ᶠ (a : α) in l, f a ∈ s
• Filter.tendsto_principal_principal 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {t : Set β} : Filter.Tendsto f (Filter.principal s) (Filter.principal t) ↔ ∀ a ∈ s, f a ∈ t
• Filter.map_inf_principal_preimage 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set β} {l : Filter α} : Filter.map f (l ⊓ Filter.principal (f ⁻¹' s)) = Filter.map f l ⊓ Filter.principal s
• Set.MapsTo.filter_map_Iic 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.MapsTo m s t → Set.MapsTo (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t))
• Filter.map_mapsTo_Iic_iff_mapsTo 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.MapsTo (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t)) ↔ Set.MapsTo m s t
• Filter.Tendsto.piecewise 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {s : Set α} [(x : α) → Decidable (x ∈ s)] (h₀ : Filter.Tendsto f (l₁ ⊓ Filter.principal s) l₂) (h₁ : Filter.Tendsto g (l₁ ⊓ Filter.principal sᶜ) l₂) : Filter.Tendsto (s.piecewise f g) l₁ l₂
• Filter.Tendsto.if 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop} [(x : α) → Decidable (p x)] (h₀ : Filter.Tendsto f (l₁ ⊓ Filter.principal {x | p x}) l₂) (h₁ : Filter.Tendsto g (l₁ ⊓ Filter.principal {x | ¬p x}) l₂) : Filter.Tendsto (fun x => if p x then f x else g x) l₁ l₂
• Filter.map_atBot_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsCodirectedOrder α] [Nonempty α] {f : α → β} : Filter.map f Filter.atBot = ⨅ a, Filter.principal (f '' {a' | a' ≤ a})
• Filter.map_atTop_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirectedOrder α] [Nonempty α] {f : α → β} : Filter.map f Filter.atTop = ⨅ a, Filter.principal (f '' {a' | a ≤ a'})
• Filter.tendsto_atBot_principal 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsCodirectedOrder α] {f : α → β} {s : Set β} : Filter.Tendsto f Filter.atBot (Filter.principal s) ↔ ∃ N, ∀ n ≤ N, f n ∈ s
• Filter.tendsto_atTop_principal 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirectedOrder α] {f : α → β} {s : Set β} : Filter.Tendsto f Filter.atTop (Filter.principal s) ↔ ∃ N, ∀ (n : α), N ≤ n → f n ∈ s
• IsMaxFilter.tendsto_principal_Iic 📋 Mathlib.Order.Filter.Extr
  {α : Type u} {β : Type v} [Preorder β] {f : α → β} {l : Filter α} {a : α} (h : IsMaxFilter f l a) : Filter.Tendsto f l (Filter.principal (Set.Iic (f a)))
• IsMinFilter.tendsto_principal_Ici 📋 Mathlib.Order.Filter.Extr
  {α : Type u} {β : Type v} [Preorder β] {f : α → β} {l : Filter α} {a : α} (h : IsMinFilter f l a) : Filter.Tendsto f l (Filter.principal (Set.Ici (f a)))
• Filter.principal_surjective 📋 Mathlib.Order.Filter.Finite
  {α : Type u} [Finite α] : Function.Surjective Filter.principal
• Filter.eq_principal_of_finite 📋 Mathlib.Order.Filter.Finite
  {α : Type u} [Finite α] (f : Filter α) : ∃ s, f = Filter.principal s
• Filter.eq_principal_of_finite_sets 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {f : Filter α} (hf : f.sets.Finite) : ∃ s, f = Filter.principal s
• Filter.iInf_principal 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Sort w} [Finite ι] (f : ι → Set α) : ⨅ i, Filter.principal (f i) = Filter.principal (⋂ i, f i)
• Filter.iInf_principal' 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type w} [Finite ι] (f : ι → Set α) : ⨅ i, Filter.principal (f i) = Filter.principal (⋂ i, f i)
• Filter.iInf_principal_finite 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type w} {s : Set ι} (hs : s.Finite) (f : ι → Set α) : ⨅ i ∈ s, Filter.principal (f i) = Filter.principal (⋂ i ∈ s, f i)
• Filter.iInf_principal_finset 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type w} (s : Finset ι) (f : ι → Set α) : ⨅ i ∈ s, Filter.principal (f i) = Filter.principal (⋂ i ∈ s, f i)
• Filter.mem_biInf_principal 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type u_1} {p : ι → Prop} {s : ι → Set α} {t : Set α} : t ∈ ⨅ i, ⨅ (_ : p i), Filter.principal (s i) ↔ ∃ I, I.Finite ∧ (∀ i ∈ I, p i) ∧ ⋂ i ∈ I, s i ⊆ t
• Filter.hasBasis_iInf_principal_finite 📋 Mathlib.Order.Filter.Bases.Finite
  {α : Type u_1} {ι : Type u_4} (s : ι → Set α) : (⨅ i, Filter.principal (s i)).HasBasis (fun t => t.Finite) fun t => ⋂ i ∈ t, s i
• Filter.isCountablyGenerated_principal 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} (s : Set α) : (Filter.principal s).IsCountablyGenerated
• Filter.isCountablyGenerated_seq 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {ι' : Sort u_4} [Countable ι'] (x : ι' → Set α) : (⨅ i, Filter.principal (x i)).IsCountablyGenerated
• Filter.isCountablyGenerated_of_seq 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {f : Filter α} (h : ∃ x, f = ⨅ i, Filter.principal (x i)) : f.IsCountablyGenerated
• Filter.isCountablyGenerated_biInf_principal 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {B : Set (Set α)} (h : B.Countable) : (⨅ s ∈ B, Filter.principal s).IsCountablyGenerated
• Filter.antitone_seq_of_seq 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} (s : ℕ → Set α) : ∃ t, Antitone t ∧ ⨅ i, Filter.principal (s i) = ⨅ i, Filter.principal (t i)
• Filter.countable_biInf_principal_eq_seq_iInf 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {B : Set (Set α)} (Bcbl : B.Countable) : ∃ x, ⨅ t ∈ B, Filter.principal t = ⨅ i, Filter.principal (x i)
• Filter.ker_principal 📋 Mathlib.Order.Filter.Ker
  {α : Type u_2} (s : Set α) : (Filter.principal s).ker = s
• Filter.giPrincipalKer 📋 Mathlib.Order.Filter.Ker
  {α : Type u_2} : GaloisCoinsertion Filter.principal Filter.ker
• Filter.gi_principal_ker 📋 Mathlib.Order.Filter.Ker
  {α : Type u_2} : GaloisCoinsertion Filter.principal Filter.ker
• Filter.hasBasis_pi_principal 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} (s : (i : ι) → Set (α i)) : (Filter.pi fun i => Filter.principal (s i)).HasBasis Set.Finite fun x => x.pi s
• Filter.pi_principal 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} [Finite ι] (s : (i : ι) → Set (α i)) : (Filter.pi fun i => Filter.principal (s i)) = Filter.principal (Set.univ.pi s)
• Filter.le_pi_principal 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} (s : (i : ι) → Set (α i)) : Filter.principal (Set.univ.pi s) ≤ Filter.pi fun i => Filter.principal (s i)
• Filter.pi_inf_principal_univ_pi_neBot 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} : (Filter.pi f ⊓ Filter.principal (Set.univ.pi s)).NeBot ↔ ∀ (i : ι), (f i ⊓ Filter.principal (s i)).NeBot
• Filter.PiInfPrincipalPi.neBot 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [h : ∀ (i : ι), (f i ⊓ Filter.principal (s i)).NeBot] {I : Set ι} : (Filter.pi f ⊓ Filter.principal (I.pi s)).NeBot
• Filter.mem_pi_principal 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {s : (i : ι) → Set (α i)} {t : Set ((i : ι) → α i)} : (t ∈ Filter.pi fun i => Filter.principal (s i)) ↔ ∃ I, I.Finite ∧ I.pi s ⊆ t
• Filter.pi_inf_principal_pi_neBot 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} : (Filter.pi f ⊓ Filter.principal (I.pi s)).NeBot ↔ ∀ i ∈ I, (f i ⊓ Filter.principal (s i)).NeBot
• Filter.pi_inf_principal_univ_pi_eq_bot 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} : Filter.pi f ⊓ Filter.principal (Set.univ.pi s) = ⊥ ↔ ∃ i, f i ⊓ Filter.principal (s i) = ⊥
• Filter.pi_inf_principal_pi_eq_bot 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} : Filter.pi f ⊓ Filter.principal (I.pi s) = ⊥ ↔ ∃ i ∈ I, f i ⊓ Filter.principal (s i) = ⊥
• Filter.prod_principal_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} : Filter.principal s ×ˢ Filter.principal t = Filter.principal (s ×ˢ t)
• Filter.eventually_prod_principal_iff 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter α} {p : α × β → Prop} {s : Set β} : (∀ᶠ (x : α × β) in f ×ˢ Filter.principal s, p x) ↔ ∀ᶠ (x : α) in f, ∀ y ∈ s, p (x, y)
• Filter.principal_coprod_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} (s : Set α) (t : Set β) : (Filter.principal s).coprod (Filter.principal t) = Filter.principal (sᶜ ×ˢ tᶜ)ᶜ
• Filter.map_prodMap_const_id_principal_coprod_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_6} {β : Type u_7} {ι : Type u_8} (a : α) (b : β) (i : ι) : Filter.map (Prod.map (fun x => b) id) ((Filter.principal {a}).coprod (Filter.principal {i})) = Filter.principal ({b} ×ˢ Set.univ)
• Filter.mem_prod_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {t : Set β} {f : Filter α} {s : Set (α × β)} : s ∈ f ×ˢ Filter.principal t ↔ {a | ∀ b ∈ t, (a, b) ∈ s} ∈ f
• Filter.Tendsto.coprod_of_prod_top_left 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α × β → γ} {la : Filter α} {lb : Filter β} {lc : Filter γ} (h₁ : ∀ s ∈ lb, Filter.Tendsto f (la ×ˢ Filter.principal sᶜ) lc) (h₂ : Filter.Tendsto f (⊤ ×ˢ lb) lc) : Filter.Tendsto f (la.coprod lb) lc
• Filter.Tendsto.coprod_of_prod_top_right 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α × β → γ} {la : Filter α} {lb : Filter β} {lc : Filter γ} (h₁ : ∀ s ∈ la, Filter.Tendsto f (Filter.principal sᶜ ×ˢ lb) lc) (h₂ : Filter.Tendsto f (la ×ˢ ⊤) lc) : Filter.Tendsto f (la.coprod lb) lc
• Filter.map_const_principal_coprod_map_id_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_6} {β : Type u_7} {ι : Type u_8} (a : α) (b : β) (i : ι) : (Filter.map (fun x => b) (Filter.principal {a})).coprod (Filter.map id (Filter.principal {i})) = Filter.principal ({b} ×ˢ Set.univ ∪ Set.univ ×ˢ {i})
• Set.Infinite.cofinite_inf_principal_neBot 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {s : Set α} : s.Infinite → (Filter.cofinite ⊓ Filter.principal s).NeBot
• Filter.cofinite_inf_principal_neBot_iff 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {s : Set α} : (Filter.cofinite ⊓ Filter.principal s).NeBot ↔ s.Infinite
• boundary_principal 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} (s : Set α) : Coheyting.boundary (Filter.principal s) = ⊥
• Set.Finite.cofinite_inf_principal_compl 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {s : Set α} (hs : s.Finite) : Filter.cofinite ⊓ Filter.principal sᶜ = Filter.cofinite
• Function.update_eventuallyEq 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {β : Type u_3} [DecidableEq α] (f : α → β) (a : α) (b : β) : Function.update f a b =ᶠ[Filter.principal {a}ᶜ] f
• Set.Finite.cofinite_inf_principal_diff 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {s t : Set α} (ht : t.Finite) : Filter.cofinite ⊓ Filter.principal (s \ t) = Filter.cofinite ⊓ Filter.principal s
• Set.Finite.cofinite_inf_principal_sdiff 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {s t : Set α} (ht : t.Finite) : Filter.cofinite ⊓ Filter.principal (s \ t) = Filter.cofinite ⊓ Filter.principal s
• exists_eq_principal_sup_free 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} (f : Filter α) : ∃ s, ∃ g ≤ Filter.cofinite, Disjoint (Filter.principal s) g ∧ f = Filter.principal s ⊔ g
• existsUnique_eq_principal_sup_free 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} (f : Filter α) : ∃! p, p.2 ≤ Filter.cofinite ∧ Disjoint (Filter.principal p.1) p.2 ∧ f = Filter.principal p.1 ⊔ p.2
• Ultrafilter.comap_inf_principal_neBot_of_image_mem 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g) : (Filter.comap m ↑g ⊓ Filter.principal s).NeBot
• Filter.principal_one 📋 Mathlib.Order.Filter.Pointwise
  {α : Type u_2} [One α] : Filter.principal 1 = 1
• Filter.principal_zero 📋 Mathlib.Order.Filter.Pointwise
  {α : Type u_2} [Zero α] : Filter.principal 0 = 0
• Filter.lift_principal2 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {f : Filter α} : f.lift Filter.principal = f
• Filter.lift'_top 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} (h : Set α → Set β) : ⊤.lift' h = Filter.principal (h Set.univ)
• Filter.lift'_inf_principal_eq 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {f : Filter α} {h : Set α → Set β} {s : Set β} : f.lift' h ⊓ Filter.principal s = f.lift' fun t => h t ∩ s
• Filter.principal_le_lift' 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {f : Filter α} {h : Set α → Set β} {t : Set β} : Filter.principal t ≤ f.lift' h ↔ ∀ s ∈ f, t ⊆ h s
• Filter.lift'_le 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Set α → Set β} {h : Filter β} {s : Set α} (hs : s ∈ f) (hg : Filter.principal (g s) ≤ h) : f.lift' g ≤ h
• Filter.lift_principal 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {g : Set α → Filter β} {s : Set α} (hg : Monotone g) : (Filter.principal s).lift g = g s
• Filter.lift'_principal 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {h : Set α → Set β} {s : Set α} (hh : Monotone h) : (Filter.principal s).lift' h = Filter.principal (h s)
• Filter.lift'_bot 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {h : Set α → Set β} (hh : Monotone h) : ⊥.lift' h = Filter.principal (h ∅)
• Filter.lift'_pure 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {h : Set α → Set β} {a : α} (hh : Monotone h) : (pure a).lift' h = Filter.principal (h {a})
• nhds_def 📋 Mathlib.Topology.Defs.Filter
  {X : Type u_3} [TopologicalSpace X] (x : X) : nhds x = ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, Filter.principal s
• interior_eq_nhds 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {s : Set X} : interior s = {x | nhds x ≤ Filter.principal s}
• isOpen_iff_nhds 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {s : Set X} : IsOpen s ↔ ∀ x ∈ s, nhds x ≤ Filter.principal s
• nhds_le_of_le 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} {f : Filter X} (h : x ∈ s) (o : IsOpen s) (sf : Filter.principal s ≤ f) : nhds x ≤ f
• tendsto_inf_principal_nhds_iff_of_forall_eq 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {x : X} {f : α → X} {l : Filter α} {s : Set α} (h : ∀ a ∉ s, f a = x) : Filter.Tendsto f (l ⊓ Filter.principal s) (nhds x) ↔ Filter.Tendsto f l (nhds x)
• nhds_def' 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] (x : X) : nhds x = ⨅ s, ⨅ (_ : IsOpen s), ⨅ (_ : x ∈ s), Filter.principal s
• map_nhds 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {x : X} {f : X → α} : Filter.map f (nhds x) = ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, Filter.principal (f '' s)
• closure_eq_cluster_pts 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {s : Set X} : closure s = {a | ClusterPt a (Filter.principal s)}
• ClusterPt.mem_closure 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : ClusterPt x (Filter.principal s) → x ∈ closure s
• mem_closure_iff_clusterPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : x ∈ closure s ↔ ClusterPt x (Filter.principal s)
• isClosed_iff_accPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {s : Set X} : IsClosed s ↔ ∀ (a : X), AccPt a (Filter.principal s) → a ∈ s
• isClosed_iff_clusterPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {s : Set X} : IsClosed s ↔ ∀ (a : X), ClusterPt a (Filter.principal s) → a ∈ s
• clusterPt_principal_iff_frequently 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : ClusterPt x (Filter.principal s) ↔ ∃ᶠ (y : X) in nhds x, y ∈ s
• clusterPt_principal 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {C : Set X} : ClusterPt x (Filter.principal C) ↔ x ∈ C ∨ AccPt x (Filter.principal C)
• mem_interior_iff_not_clusterPt_compl 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : x ∈ interior s ↔ ¬ClusterPt x (Filter.principal sᶜ)
• accPt_principal_iff_clusterPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {C : Set X} : AccPt x (Filter.principal C) ↔ ClusterPt x (Filter.principal (C \ {x}))
• accPt_principal_iff_nhdsWithin 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : AccPt x (Filter.principal s) ↔ (nhdsWithin x (s \ {x})).NeBot
• accPt_iff_frequently 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {C : Set X} : AccPt x (Filter.principal C) ↔ ∃ᶠ (y : X) in nhds x, y ≠ x ∧ y ∈ C
• accPt_iff_clusterPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {F : Filter X} : AccPt x F ↔ ClusterPt x (Filter.principal {x}ᶜ ⊓ F)
• clusterPt_principal_iff 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : ClusterPt x (Filter.principal s) ↔ ∀ U ∈ nhds x, (U ∩ s).Nonempty
• mem_closure_iff_nhds_ne_bot 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : x ∈ closure s ↔ nhds x ⊓ Filter.principal s ≠ ⊥
• accPt_iff_frequently_nhdsNE 📋 Mathlib.Topology.ClusterPt
  {X : Type u_3} [TopologicalSpace X] {x : X} {C : Set X} : AccPt x (Filter.principal C) ↔ ∃ᶠ (y : X) in nhdsWithin x {x}ᶜ, y ∈ C
• accPt_iff_nhds 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {C : Set X} : AccPt x (Filter.principal C) ↔ ∀ U ∈ nhds x, ∃ y ∈ U ∩ C, y ≠ x
• isClosed_iff_forall_filter 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {s : Set X} : IsClosed s ↔ ∀ (x : X) (F : Filter X), F.NeBot → F ≤ Filter.principal s → F ≤ nhds x → x ∈ s
• TopologicalSpace.nhds_generateFrom 📋 Mathlib.Topology.Order
  {α : Type u} {g : Set (Set α)} {a : α} : nhds a = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, Filter.principal s
• IsOpen.nhdsSet_eq 📋 Mathlib.Topology.NhdsSet
  {X : Type u_2} [TopologicalSpace X] {s : Set X} : IsOpen s → nhdsSet s = Filter.principal s
• nhdsSet_eq_principal_iff 📋 Mathlib.Topology.NhdsSet
  {X : Type u_2} [TopologicalSpace X] {s : Set X} : nhdsSet s = Filter.principal s ↔ IsOpen s

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59