Loogle!

Result

Found 2450 declarations mentioning UniformSpace. Of these, only the first 200 are shown.

• UniformSpace 📋 Mathlib.Topology.UniformSpace.Defs
  (α : Type u) : Type u
• UniformSpace.ofCore 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} (u : UniformSpace.Core α) : UniformSpace α
• UniformSpace.toCore 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} (u : UniformSpace α) : UniformSpace.Core α
• UniformSpace.toTopologicalSpace 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] : TopologicalSpace α
• uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  (α : Type u) [UniformSpace α] : Filter (α × α)
• UniformSpace.uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] : Filter (α × α)
• IsUniformEmbedding 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) : Prop
• IsUniformInducing 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) : Prop
• UniformContinuous 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) : Prop
• uniformContinuous_id 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : UniformContinuous id
• UniformContinuousOn 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) (s : Set α) : Prop
• uniformity.neBot 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] [Nonempty α] : (uniformity α).NeBot
• uniformContinuous_const 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {b : β} : UniformContinuous fun x => b
• UniformSpace.ofCoreEq 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} (u : UniformSpace.Core α) (t : TopologicalSpace α) (h : t = u.toTopologicalSpace) : UniformSpace α
• UniformSpace.replaceTopology 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u_2} [i : TopologicalSpace α] (u : UniformSpace α) (h : i = u.toTopologicalSpace) : UniformSpace α
• UniformSpace.toCore_toTopologicalSpace 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} (u : UniformSpace α) : u.toCore.toTopologicalSpace = u.toTopologicalSpace
• IsUniformEmbedding.injective 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (self : IsUniformEmbedding f) : Function.Injective f
• tendsto_swap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.Tendsto Prod.swap (uniformity α) (uniformity α)
• UniformSpace.symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] : Filter.Tendsto Prod.swap UniformSpace.uniformity UniformSpace.uniformity
• IsUniformEmbedding.isUniformInducing 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : IsUniformEmbedding f) : IsUniformInducing f
• IsUniformEmbedding.toIsUniformInducing 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (self : IsUniformEmbedding f) : IsUniformInducing f
• UniformContinuous.iterate 📋 Mathlib.Topology.UniformSpace.Defs
  {β : Type ub} [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T) : UniformContinuous T^[n]
• UniformSpace.ext 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {u₁ u₂ : UniformSpace α} (h : uniformity α = uniformity α) : u₁ = u₂
• UniformSpace.replaceTopology_eq 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u_2} [i : TopologicalSpace α] (u : UniformSpace α) (h : i = u.toTopologicalSpace) : u.replaceTopology h = u
• tendsto_const_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {a : α} {f : Filter β} : Filter.Tendsto (fun x => (a, a)) f (uniformity α)
• uniformContinuous_of_const 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {c : α → β} (h : ∀ (a b : α), c a = c b) : UniformContinuous c
• uniformContinuousOn_univ 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} : UniformContinuousOn f Set.univ ↔ UniformContinuous f
• comap_swap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.comap Prod.swap (uniformity α) = uniformity α
• nhds_eq_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} : nhds x = (uniformity α).lift' (UniformSpace.ball x)
• tendsto_left_nhds_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {a : α} : Filter.Tendsto (fun a' => (a, a')) (nhds a) (uniformity α)
• tendsto_right_nhds_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {a : α} : Filter.Tendsto (fun a' => (a', a)) (nhds a) (uniformity α)
• uniformity_eq_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : uniformity α = Filter.map Prod.swap (uniformity α)
• IsUniformEmbedding.mk 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (toIsUniformInducing : IsUniformInducing f) (injective : Function.Injective f) : IsUniformEmbedding f
• isRefl_of_mem_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (h : s ∈ uniformity α) : s.IsRefl
• nhds_eq_comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} : nhds x = Filter.comap (Prod.mk x) (uniformity α)
• tendsto_diag_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] (f : β → α) (l : Filter β) : Filter.Tendsto (fun x => (f x, f x)) l (uniformity α)
• UniformSpace.nhds_eq_comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] (x : α) : nhds x = Filter.comap (Prod.mk x) UniformSpace.uniformity
• UniformSpace.ofCoreEq_toCore 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} (u : UniformSpace α) (t : TopologicalSpace α) (h : t = u.toCore.toTopologicalSpace) : UniformSpace.ofCoreEq u.toCore t h = u
• isUniformEmbedding_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) : IsUniformEmbedding f ↔ IsUniformInducing f ∧ Function.Injective f
• nhds_eq_comap_uniformity' 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} : nhds x = Filter.comap (fun y => (y, x)) (uniformity α)
• lift'_comp_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : ((uniformity α).lift' fun s => s.comp s) = uniformity α
• refl_le_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.principal SetRel.id ≤ uniformity α
• UniformSpace.mem_ball_self 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] (x : α) {V : SetRel α α} : V ∈ uniformity α → x ∈ UniformSpace.ball x V
• UniformContinuous.comp 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f) : UniformContinuous (g ∘ f)
• nhds_basis_uniformity' 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {s : ι → SetRel α α} (h : (uniformity α).HasBasis p s) {x : α} : (nhds x).HasBasis p fun i => UniformSpace.ball x (s i)
• subset_comp_self_of_mem_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (h : s ∈ uniformity α) : s ⊆ s.comp s
• symm_le_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.map Prod.swap (uniformity α) ≤ uniformity α
• uniformity_le_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : uniformity α ≤ Filter.map Prod.swap (uniformity α)
• refl_mem_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {s : SetRel α α} (h : s ∈ uniformity α) : (x, x) ∈ s
• symmetrize_mem_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {V : SetRel α α} (h : V ∈ uniformity α) : V.symmetrize ∈ uniformity α
• UniformSpace.ball_mem_nhds 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] (x : α) ⦃V : SetRel α α⦄ (V_in : V ∈ uniformity α) : UniformSpace.ball x V ∈ nhds x
• UniformSpace.closure_subset_image 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {U : SetRel α α} (hU : U ∈ uniformity α) (s : Set α) : closure s ⊆ U.image s
• UniformSpace.closure_subset_preimage 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {U : SetRel α α} (hU : U ∈ uniformity α) (s : Set α) : closure s ⊆ U.preimage s
• UniformSpace.hasBasis_symmetric 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : (uniformity α).HasBasis (fun s => s ∈ uniformity α ∧ s.IsSymm) id
• Filter.Tendsto.uniformity_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {l : Filter β} {f : β → α × α} (h : Filter.Tendsto f l (uniformity α)) : Filter.Tendsto (fun x => ((f x).2, (f x).1)) l (uniformity α)
• comp_le_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : ((uniformity α).lift' fun s => s.comp s) ≤ uniformity α
• mem_uniformity_of_eq 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x y : α} {s : SetRel α α} (h : s ∈ uniformity α) (hx : x = y) : (x, y) ∈ s
• UniformSpace.comp 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] : (UniformSpace.uniformity.lift' fun s => SetRel.comp s s) ≤ UniformSpace.uniformity
• UniformSpace.hasBasis_nhds 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] (x : α) : (nhds x).HasBasis (fun s => s ∈ uniformity α ∧ s.IsSymm) fun s => UniformSpace.ball x s
• nhds_eq_uniformity' 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} : nhds x = (uniformity α).lift' fun s => {y | (y, x) ∈ s}
• IsUniformInducing.comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (self : IsUniformInducing f) : Filter.comap (fun x => (f x.1, f x.2)) (uniformity β) = uniformity α
• IsUniformInducing.mk 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (comap_uniformity : Filter.comap (fun x => (f x.1, f x.2)) (uniformity β) = uniformity α) : IsUniformInducing f
• comp_le_uniformity3 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : ((uniformity α).lift' fun s => s.comp (s.comp s)) ≤ uniformity α
• isUniformInducing_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] (f : α → β) : IsUniformInducing f ↔ Filter.comap (fun x => (f x.1, f x.2)) (uniformity β) = uniformity α
• UniformSpace.ext_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {u₁ u₂ : UniformSpace α} : u₁ = u₂ ↔ ∀ (s : Set (α × α)), s ∈ uniformity α ↔ s ∈ uniformity α
• nhds_basis_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {s : ι → SetRel α α} (h : (uniformity α).HasBasis p s) {x : α} : (nhds x).HasBasis p fun i => {y | (y, x) ∈ s i}
• UniformSpace.mem_closure_iff_ball 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : Set α} {x : α} : x ∈ closure s ↔ ∀ {V : Set (α × α)}, V ∈ uniformity α → (UniformSpace.ball x V ∩ s).Nonempty
• mem_nhds_left 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] (x : α) {s : SetRel α α} (h : s ∈ uniformity α) : {y | (x, y) ∈ s} ∈ nhds x
• mem_nhds_right 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] (y : α) {s : SetRel α α} (h : s ∈ uniformity α) : {x | (x, y) ∈ s} ∈ nhds y
• nhdsWithin_eq_comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} (S : Set α) : nhdsWithin x S = Filter.comap (Prod.mk x) (uniformity α ⊓ Filter.principal (Set.univ ×ˢ S))
• UniformSpace.mem_closure_iff_symm_ball 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : Set α} {x : α} : x ∈ closure s ↔ ∀ {V : Set (α × α)}, V ∈ uniformity α → SetRel.IsSymm V → (s ∩ UniformSpace.ball x V).Nonempty
• isOpen_iff_ball_subset 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : Set α} : IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ uniformity α, UniformSpace.ball x V ⊆ s
• UniformSpace.mem_nhds_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {s : Set α} : s ∈ nhds x ↔ ∃ V ∈ uniformity α, UniformSpace.ball x V ⊆ s
• isOpen_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : Set α} : IsOpen s ↔ ∀ x ∈ s, {p | p.1 = x → p.2 ∈ s} ∈ uniformity α
• mem_nhds_uniformity_iff_left 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {s : Set α} : s ∈ nhds x ↔ {p | p.2 = x → p.1 ∈ s} ∈ uniformity α
• mem_nhds_uniformity_iff_right 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {s : Set α} : s ∈ nhds x ↔ {p | p.1 = x → p.2 ∈ s} ∈ uniformity α
• UniformSpace.mem_nhds_iff_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {s : Set α} : s ∈ nhds x ↔ ∃ V ∈ uniformity α, SetRel.IsSymm V ∧ UniformSpace.ball x V ⊆ s
• Filter.Tendsto.uniformity_trans 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {l : Filter β} {f₁ f₂ f₃ : β → α} (h₁₂ : Filter.Tendsto (fun x => (f₁ x, f₂ x)) l (uniformity α)) (h₂₃ : Filter.Tendsto (fun x => (f₂ x, f₃ x)) l (uniformity α)) : Filter.Tendsto (fun x => (f₁ x, f₃ x)) l (uniformity α)
• comp_mem_uniformity_sets 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, SetRel.comp t t ⊆ s
• symm_of_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, SetRel.IsSymm t ∧ t ⊆ s
• nhdsWithin_eq_comap_uniformity_of_mem 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {T : Set α} (hx : x ∈ T) (S : Set α) : nhdsWithin x S = Filter.comap (Prod.mk x) (uniformity α ⊓ Filter.principal (T ×ˢ S))
• comp_symm_mem_uniformity_sets 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, SetRel.IsSymm t ∧ SetRel.comp t t ⊆ s
• Filter.HasBasis.biInter_biUnion_ball 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {U : ι → SetRel α α} (h : (uniformity α).HasBasis p U) (s : Set α) : ⋂ i, ⋂ (_ : p i), ⋃ x ∈ s, UniformSpace.ball x (U i) = closure s
• comp3_mem_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, SetRel.comp t (SetRel.comp t t) ⊆ s
• lift_nhds_left 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {x : α} {g : Set α → Filter β} (hg : Monotone g) : (nhds x).lift g = (uniformity α).lift fun s => g (UniformSpace.ball x s)
• uniformContinuous_iff_eventually 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} : UniformContinuous f ↔ ∀ r ∈ uniformity β, ∀ᶠ (x : α × α) in uniformity α, (f x.1, f x.2) ∈ r
• comp_comp_symm_mem_uniformity_sets 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, SetRel.IsSymm t ∧ (SetRel.comp t t).comp t ⊆ s
• UniformSpace.mk 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [toTopologicalSpace : TopologicalSpace α] (uniformity : Filter (α × α)) (symm : Filter.Tendsto Prod.swap uniformity uniformity) (comp : (uniformity.lift' fun s => SetRel.comp s s) ≤ uniformity) (nhds_eq_comap_uniformity : ∀ (x : α), nhds x = Filter.comap (Prod.mk x) uniformity) : UniformSpace α
• UniformSpace.ball_mem_nhdsWithin 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} {S : Set α} ⦃V : SetRel α α⦄ (x_in : x ∈ S) (V_in : V ∈ uniformity α ⊓ Filter.principal (S ×ˢ S)) : UniformSpace.ball x V ∈ nhdsWithin x S
• uniformContinuous_def 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} : UniformContinuous f ↔ ∀ r ∈ uniformity β, {x | (f x.1, f x.2) ∈ r} ∈ uniformity α
• exists_mem_nhds_ball_subset_of_mem_nhds 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {a : α} {U : Set α} (h : U ∈ nhds a) : ∃ V ∈ nhds a, ∃ t ∈ uniformity α, ∀ a' ∈ V, UniformSpace.ball a' t ⊆ U
• lift_nhds_right 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {x : α} {g : Set α → Filter β} (hg : Monotone g) : (nhds x).lift g = (uniformity α).lift fun s => g {y | (y, x) ∈ s}
• uniformity_lift_le_comp 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {f : SetRel α α → Filter β} (h : Monotone f) : ((uniformity α).lift fun s => f (SetRel.comp s s)) ≤ (uniformity α).lift f
• Filter.HasBasis.uniformContinuous_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} {ι : Sort u_1} [UniformSpace α] [UniformSpace β] {ι' : Sort u_2} {p : ι → Prop} {s : ι → SetRel α α} (ha : (uniformity α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)} (hb : (uniformity β).HasBasis q t) {f : α → β} : UniformContinuous f ↔ ∀ (i : ι'), q i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ t i
• comp_symm_of_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {s : SetRel α α} (hs : s ∈ uniformity α) : ∃ t ∈ uniformity α, (∀ {a b : α}, (a, b) ∈ t → (b, a) ∈ t) ∧ SetRel.comp t t ⊆ s
• uniformity_lift_le_swap 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] {g : SetRel α α → Filter β} {f : Filter β} (hg : Monotone g) (h : ((uniformity α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f) : (uniformity α).lift g ≤ f
• nhds_nhds_eq_uniformity_uniformity_prod 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {a b : α} : nhds a ×ˢ nhds b = (uniformity α).lift fun s => (uniformity α).lift' fun t => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}
• Filter.HasBasis.uniformContinuousOn_iff 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} {ι : Sort u_1} [UniformSpace α] [UniformSpace β] {ι' : Sort u_2} {p : ι → Prop} {s : ι → SetRel α α} (ha : (uniformity α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)} (hb : (uniformity β).HasBasis q t) {f : α → β} {S : Set α} : UniformContinuousOn f S ↔ ∀ (i : ι'), q i → ∃ j, p j ∧ ∀ x ∈ S, ∀ y ∈ S, (x, y) ∈ s j → (f x, f y) ∈ t i
• instUniformSpaceBool 📋 Mathlib.Topology.UniformSpace.Basic
  : UniformSpace Bool
• instUniformSpaceEmpty 📋 Mathlib.Topology.UniformSpace.Basic
  : UniformSpace Empty
• instUniformSpaceInt 📋 Mathlib.Topology.UniformSpace.Basic
  : UniformSpace ℤ
• instUniformSpaceNat 📋 Mathlib.Topology.UniformSpace.Basic
  : UniformSpace ℕ
• instUniformSpacePUnit 📋 Mathlib.Topology.UniformSpace.Basic
  : UniformSpace PUnit.{u_2 + 1}
• inhabitedUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : Inhabited (UniformSpace α)
• instBotUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : Bot (UniformSpace α)
• instCompleteLatticeUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : CompleteLattice (UniformSpace α)
• instInfSetUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : InfSet (UniformSpace α)
• instMinUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : Min (UniformSpace α)
• instPartialOrderUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : PartialOrder (UniformSpace α)
• instTopUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : Top (UniformSpace α)
• instUniformSpaceAddOpposite 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace αᵃᵒᵖ
• instUniformSpaceAdditive 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace (Additive α)
• instUniformSpaceMulOpposite 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace αᵐᵒᵖ
• instUniformSpaceMultiplicative 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace (Multiplicative α)
• instUniqueUniformSpaceOfSubsingleton 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [Subsingleton α] : Unique (UniformSpace α)
• OrderDual.instUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace αᵒᵈ
• ULift.uniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformSpace (ULift.{u_2, ua} α)
• UniformSpace.comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} (f : α → β) (u : UniformSpace β) : UniformSpace α
• instUniformSpaceSubtype 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {p : α → Prop} [t : UniformSpace α] : UniformSpace (Subtype p)
• instUniformSpaceProd 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [u₁ : UniformSpace α] [u₂ : UniformSpace β] : UniformSpace (α × β)
• Sum.instUniformSpace 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformSpace (α ⊕ β)
• UniformContinuous₂ 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (f : α → β → γ) : Prop
• AddOpposite.uniformContinuous_op 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous AddOpposite.op
• AddOpposite.uniformContinuous_unop 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous AddOpposite.unop
• MulOpposite.uniformContinuous_op 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous MulOpposite.op
• MulOpposite.uniformContinuous_unop 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous MulOpposite.unop
• uniformSpace_comap_id 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} : UniformSpace.comap id = id
• uniformContinuous_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} [u : UniformSpace β] : UniformContinuous f
• bot_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : uniformity α = Filter.principal SetRel.id
• uniformContinuous_inl 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformContinuous Sum.inl
• uniformContinuous_inr 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformContinuous Sum.inr
• discreteTopology_of_discrete_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [hα : UniformSpace α] (h : uniformity α = Filter.principal SetRel.id) : DiscreteTopology α
• top_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : uniformity α = ⊤
• UniformSpace.isGLB_sInf 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {tt : Set (UniformSpace α)} : IsGLB tt (sInf tt)
• UniformContinuous.uniformContinuousOn 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} (hf : UniformContinuous f) : UniformContinuousOn f s
• UniformSpace.toTopologicalSpace_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u : UniformSpace β} : (UniformSpace.comap f u).toTopologicalSpace = TopologicalSpace.induced f u.toTopologicalSpace
• toTopologicalSpace_subtype 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [u : UniformSpace α] {p : α → Prop} : instUniformSpaceSubtype.toTopologicalSpace = instTopologicalSpaceSubtype
• uniformContinuous_fst 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformContinuous fun p => p.1
• uniformContinuous_snd 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformContinuous fun p => p.2
• uniformContinuous_subtype_val 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {p : α → Prop} [UniformSpace α] : UniformContinuous Subtype.val
• UniformContinuous.continuous 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformContinuous f) : Continuous f
• le_iff_uniformContinuous_id 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u v : UniformSpace α} : u ≤ v ↔ UniformContinuous id
• uniformContinuous_swap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] : UniformContinuous Prod.swap
• UniformSpace.comap_mono 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {γ : Type u_3} {f : α → γ} : Monotone fun u => UniformSpace.comap f u
• UniformContinuous.inf_dom_left 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β} (hf : UniformContinuous f) : UniformContinuous f
• UniformContinuous.inf_dom_right 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β} (hf : UniformContinuous f) : UniformContinuous f
• UniformContinuousOn.continuousOn 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} (h : UniformContinuousOn f s) : ContinuousOn f s
• toTopologicalSpace_prod 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} [u : UniformSpace α] [v : UniformSpace β] : instUniformSpaceProd.toTopologicalSpace = instTopologicalSpaceProd
• uniformContinuous_iInf_dom 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {ι : Sort u_1} {f : α → β} {u₁ : ι → UniformSpace α} {u₂ : UniformSpace β} {i : ι} (hf : UniformContinuous f) : UniformContinuous f
• UniformSpace.isClosed_ball 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] (x : α) {V : SetRel α α} (hV : IsClosed V) : IsClosed (UniformSpace.ball x V)
• UniformSpace.isOpen_ball 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] (x : α) {V : SetRel α α} (hV : IsOpen V) : IsOpen (UniformSpace.ball x V)
• UniformSpace.uniformSpace_eq_bot 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u : UniformSpace α} : u = ⊥ ↔ SetRel.id ∈ uniformity α
• uniformContinuous_iInf_rng 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {ι : Sort u_1} {f : α → β} {u₁ : UniformSpace α} {u₂ : ι → UniformSpace β} : UniformContinuous f ↔ ∀ (i : ι), UniformContinuous f
• UniformSpace.comap_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {uγ : UniformSpace γ} {f : α → β} {g : β → γ} : UniformSpace.comap (g ∘ f) uγ = UniformSpace.comap f (UniformSpace.comap g uγ)
• uniformContinuous_comap' 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} {f : γ → β} {g : α → γ} [v : UniformSpace β] [u : UniformSpace α] (h : UniformContinuous (f ∘ g)) : UniformContinuous g
• uniformContinuous_iff 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {uα : UniformSpace α} {uβ : UniformSpace β} {f : α → β} : UniformContinuous f ↔ uα ≤ UniformSpace.comap f uβ
• uniformContinuous_iff_le_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {uα : UniformSpace α} {uβ : UniformSpace β} {f : α → β} : UniformContinuous f ↔ uα ≤ UniformSpace.comap f uβ
• UniformContinuousOn.congr 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f g : α → β} {s : Set α} (hf : UniformContinuousOn f s) (h : Set.EqOn f g s) : UniformContinuousOn g s
• UniformContinuousOn.mono 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s t : Set α} (hf : UniformContinuousOn f s) (ht : t ⊆ s) : UniformContinuousOn f t
• iInf_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {ι : Sort u_2} {u : ι → UniformSpace α} : uniformity α = ⨅ i, uniformity α
• inf_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u v : UniformSpace α} : uniformity α = uniformity α ⊓ uniformity α
• uniformContinuous_ofAdd 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous ⇑Multiplicative.ofAdd
• uniformContinuous_ofMul 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous ⇑Additive.ofMul
• uniformity_eq_uniformity_closure 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : uniformity α = (uniformity α).lift' closure
• uniformity_eq_uniformity_interior 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : uniformity α = (uniformity α).lift' interior
• UniformContinuous.inf_rng 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u₁ : UniformSpace α} {u₂ u₃ : UniformSpace β} (h₁ : UniformContinuous f) (h₂ : UniformContinuous f) : UniformContinuous f
• uniformContinuous_toAdd 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous ⇑Multiplicative.toAdd
• uniformContinuous_toMul 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : UniformContinuous ⇑Additive.toMul
• uniformity_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {x✝ : UniformSpace β} (f : α → β) : uniformity α = Filter.comap (Prod.map f f) (uniformity β)
• UniformSpace.has_seq_basis 📋 Mathlib.Topology.UniformSpace.Basic
  (α : Type ua) [UniformSpace α] [(uniformity α).IsCountablyGenerated] : ∃ V, (uniformity α).HasAntitoneBasis V ∧ ∀ (n : ℕ), (V n).IsSymm
• UniformSpace.sInf_le 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {tt : Set (UniformSpace α)} {t : UniformSpace α} (h : t ∈ tt) : sInf tt ≤ t
• UniformSpace.toTopologicalSpace_mono 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u₁ u₂ : UniformSpace α} (h : u₁ ≤ u₂) : u₁.toTopologicalSpace ≤ u₂.toTopologicalSpace
• instIsCountablyGeneratedProdElemUniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] [(uniformity α).IsCountablyGenerated] (s : Set α) : (uniformity ↑s).IsCountablyGenerated
• UniformContinuous₂.uniformContinuous 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β → γ} (h : UniformContinuous₂ f) : UniformContinuous (Function.uncurry f)
• instIsCountablyGeneratedProdSumUniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] [(uniformity α).IsCountablyGenerated] [(uniformity β).IsCountablyGenerated] : (uniformity (α ⊕ β)).IsCountablyGenerated
• instIsCountablyGeneratedProdUniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [(uniformity α).IsCountablyGenerated] [UniformSpace β] [(uniformity β).IsCountablyGenerated] : (uniformity (α × β)).IsCountablyGenerated
• uniformContinuous₂_def 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (f : α → β → γ) : UniformContinuous₂ f ↔ UniformContinuous (Function.uncurry f)
• Uniform.tendsto_nhds_left 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] {f : Filter β} {u : β → α} {a : α} : Filter.Tendsto u f (nhds a) ↔ Filter.Tendsto (fun x => (u x, a)) f (uniformity α)
• Uniform.tendsto_nhds_right 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] {f : Filter β} {u : β → α} {a : α} : Filter.Tendsto u f (nhds a) ↔ Filter.Tendsto (fun x => (a, u x)) f (uniformity α)
• UniformSpace.comap_iInf 📋 Mathlib.Topology.UniformSpace.Basic
  {ι : Sort u_2} {α : Type u_3} {γ : Type u_4} {u : ι → UniformSpace γ} {f : α → γ} : UniformSpace.comap f (⨅ i, u i) = ⨅ i, UniformSpace.comap f (u i)
• UniformSpace.comap_inf 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {γ : Type u_3} {u₁ u₂ : UniformSpace γ} {f : α → γ} : UniformSpace.comap f (u₁ ⊓ u₂) = UniformSpace.comap f u₁ ⊓ UniformSpace.comap f u₂
• closure_ball_subset 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] {x : α} {V : SetRel α α} : closure (UniformSpace.ball x V) ⊆ UniformSpace.ball x (closure V)
• uniformContinuous₂_curry 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (f : α × β → γ) : UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f
• Uniform.continuous_iff'_left 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ (b : β), Filter.Tendsto (fun x => (f x, f b)) (nhds b) (uniformity α)
• Uniform.continuous_iff'_right 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ (b : β), Filter.Tendsto (fun x => (f b, f x)) (nhds b) (uniformity α)
• UniformContinuous.prodMk_left 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α × β → γ} (h : UniformContinuous f) (b : β) : UniformContinuous fun a => f (a, b)
• UniformContinuous.prodMk_right 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α × β → γ} (h : UniformContinuous f) (a : α) : UniformContinuous fun b => f (a, b)
• UniformContinuous.subtype_mk 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {p : α → Prop} [UniformSpace α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (h : ∀ (x : β), p (f x)) : UniformContinuous fun x => ⟨f x, ⋯⟩
• uniformContinuous_sInf_dom 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u₁ : Set (UniformSpace α)} {u₂ : UniformSpace β} {u : UniformSpace α} (h₁ : u ∈ u₁) (hf : UniformContinuous f) : UniformContinuous f
• Uniform.continuousAt_iff'_left 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ Filter.Tendsto (fun x => (f x, f b)) (nhds b) (uniformity α)
• Uniform.continuousAt_iff'_right 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ Filter.Tendsto (fun x => (f b, f x)) (nhds b) (uniformity α)
• nhdsSet_diagonal_le_uniformity 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] : nhdsSet (Set.diagonal α) ≤ uniformity α
• uniformContinuous_sInf_rng 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} {u₁ : UniformSpace α} {u₂ : Set (UniformSpace β)} : UniformContinuous f ↔ ∀ u ∈ u₂, UniformContinuous f
• UniformContinuous.comp_uniformContinuousOn 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {s : Set α} {g : β → γ} (hg : UniformContinuous g) (hf : UniformContinuousOn f s) : UniformContinuousOn (g ∘ f) s
• UniformSpace.toTopologicalSpace_iInf 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {ι : Sort u_2} {u : ι → UniformSpace α} : (iInf u).toTopologicalSpace = ⨅ i, (u i).toTopologicalSpace
• UniformSpace.toTopologicalSpace_inf 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u v : UniformSpace α} : (u ⊓ v).toTopologicalSpace = u.toTopologicalSpace ⊓ v.toTopologicalSpace
• UniformContinuousOn.of_restrict 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} : UniformContinuous (s.domRestrict f) → UniformContinuousOn f s
• UniformContinuousOn.restrict 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s → UniformContinuous (s.domRestrict f)
• uniformContinuousOn_iff_restrict 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ UniformContinuous (s.domRestrict f)
• UniformSpace.to_nhds_mono 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {u₁ u₂ : UniformSpace α} (h : u₁ ≤ u₂) (a : α) : nhds a ≤ nhds a

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