Loogle!

Result

Found 411 declarations mentioning UniformContinuous. Of these, only the first 200 are shown.

• 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
• uniformContinuous_const 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {b : β} : UniformContinuous fun x => b
• UniformContinuous.iterate 📋 Mathlib.Topology.UniformSpace.Defs
  {β : Type ub} [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T) : UniformContinuous T^[n]
• 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
• 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)
• 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
• 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
• 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
• 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
• uniformContinuous_comap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {f : α → β} [u : UniformSpace β] : UniformContinuous f
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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)
• 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)
• 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
• 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
• 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
• 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)
• UniformContinuous.prodMk 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f₁ : α → β} {f₂ : α → γ} (h₁ : UniformContinuous f₁) (h₂ : UniformContinuous f₂) : UniformContinuous fun a => (f₁ a, f₂ a)
• UniformContinuous₂.comp 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] {f : α → β → γ} {g : γ → δ} (hg : UniformContinuous g) (hf : UniformContinuous₂ f) : UniformContinuous₂ (Function.bicompr g f)
• UniformContinuous.subtype_map 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {p : α → Prop} {q : β → Prop} {f : α → β} (hf : UniformContinuous f) (h : ∀ (x : α), p x → q (f x)) : UniformContinuous (Subtype.map f h)
• UniformContinuous.prodMap 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] {f : α → γ} {g : β → δ} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous (Prod.map f g)
• UniformContinuous₂.bicompl 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} {δ' : Type u_2} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] [UniformSpace δ'] {f : α → β → γ} {ga : δ → α} {gb : δ' → β} (hf : UniformContinuous₂ f) (hga : UniformContinuous ga) (hgb : UniformContinuous gb) : UniformContinuous₂ (Function.bicompl f ga gb)
• uniformContinuous_inf_dom_left₂ 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {f : α → β → γ} {ua1 ua2 : UniformSpace α} {ub1 ub2 : UniformSpace β} {uc1 : UniformSpace γ} (h : UniformContinuous fun p => f p.1 p.2) : UniformContinuous fun p => f p.1 p.2
• uniformContinuous_inf_dom_right₂ 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {f : α → β → γ} {ua1 ua2 : UniformSpace α} {ub1 ub2 : UniformSpace β} {uc1 : UniformSpace γ} (h : UniformContinuous fun p => f p.1 p.2) : UniformContinuous fun p => f p.1 p.2
• tendsto_of_uniformContinuous_subtype 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} {a : α} (hf : UniformContinuous fun x => f ↑x) (ha : s ∈ nhds a) : Filter.Tendsto f (nhds a) (nhds (f a))
• uniformContinuous_sInf_dom₂ 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {f : α → β → γ} {uas : Set (UniformSpace α)} {ubs : Set (UniformSpace β)} {ua : UniformSpace α} {ub : UniformSpace β} {uc : UniformSpace γ} (ha : ua ∈ uas) (hb : ub ∈ ubs) (hf : UniformContinuous fun p => f p.1 p.2) : UniformContinuous fun p => f p.1 p.2
• mem_uniformity_of_uniformContinuous_invariant 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {s : SetRel β β} {f : α → α → β} (hf : UniformContinuous fun p => f p.1 p.2) (hs : s ∈ uniformity β) : ∃ u ∈ uniformity α, ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s
• DiscreteUniformity.uniformContinuous 📋 Mathlib.Topology.UniformSpace.DiscreteUniformity
  (X : Type u_1) [u : UniformSpace X] [DiscreteUniformity X] {Y : Type u_2} [UniformSpace Y] (f : X → Y) : UniformContinuous f
• Cauchy.map 📋 Mathlib.Topology.UniformSpace.Cauchy
  {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : Filter α} {m : α → β} (hf : Cauchy f) (hm : UniformContinuous m) : Cauchy (Filter.map m f)
• TotallyBounded.image 📋 Mathlib.Topology.UniformSpace.Cauchy
  {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s) (hf : UniformContinuous f) : TotallyBounded (f '' s)
• Filter.TotallyBounded.map 📋 Mathlib.Topology.UniformSpace.Cauchy
  {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : α → β} {g : Filter α} (hg : g.TotallyBounded) (hf : UniformContinuous f) : (Filter.map f g).TotallyBounded
• UniformContinuous.comp_cauchySeq 📋 Mathlib.Topology.UniformSpace.Cauchy
  {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Type u_1} [UniformSpace β] [Preorder γ] {f : α → β} (hf : UniformContinuous f) {u : γ → α} (hu : CauchySeq u) : CauchySeq (f ∘ u)
• UniformContinuous.comp_uniformCauchySeqOn 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (hf : UniformCauchySeqOn F p s) : UniformCauchySeqOn (fun n => g ∘ F n) p s
• UniformContinuous.comp_tendstoUniformly 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformly F f p) : TendstoUniformly (fun i => g ∘ F i) (g ∘ f) p
• UniformContinuous.comp_tendstoUniformlyOn 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformlyOn F f p s) : TendstoUniformlyOn (fun i => g ∘ F i) (g ∘ f) p s
• UniformContinuous.comp_tendstoUniformlyOnFilter 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformlyOnFilter F f p p') : TendstoUniformlyOnFilter (fun i => g ∘ F i) (g ∘ f) p p'
• IndiscreteTopology.uniformContinuous 📋 Mathlib.Topology.UniformSpace.Separation
  {β : Type v} [UniformSpace β] {α : Type u_1} [u : UniformSpace α] [IndiscreteTopology β] {f : α → β} : UniformContinuous f
• SeparationQuotient.uniformContinuous_mk 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] : UniformContinuous SeparationQuotient.mk
• SeparationQuotient.uniformContinuous_lift' 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space β] (f : α → β) : UniformContinuous (SeparationQuotient.lift' f)
• SeparationQuotient.uniformContinuous_map 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] (f : α → β) : UniformContinuous (SeparationQuotient.map f)
• SeparationQuotient.lift'_mk 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space β] {f : α → β} (h : UniformContinuous f) (a : α) : SeparationQuotient.lift' f (SeparationQuotient.mk a) = f a
• SeparationQuotient.map_mk 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : UniformContinuous f) (a : α) : SeparationQuotient.map f (SeparationQuotient.mk a) = SeparationQuotient.mk (f a)
• SeparationQuotient.uniformContinuous_dom 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : SeparationQuotient α → β} : UniformContinuous f ↔ UniformContinuous (f ∘ SeparationQuotient.mk)
• SeparationQuotient.uniformContinuous_lift 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : ∀ (a b : α), Inseparable a b → f a = f b) : UniformContinuous (SeparationQuotient.lift f h) ↔ UniformContinuous f
• SeparationQuotient.map_comp 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) : SeparationQuotient.map g ∘ SeparationQuotient.map f = SeparationQuotient.map (g ∘ f)
• SeparationQuotient.map_unique 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformContinuous f) {g : SeparationQuotient α → SeparationQuotient β} (comm : SeparationQuotient.mk ∘ f = g ∘ SeparationQuotient.mk) : SeparationQuotient.map f = g
• SeparationQuotient.uniformContinuous_dom₂ 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : SeparationQuotient α × SeparationQuotient β → γ} : UniformContinuous f ↔ UniformContinuous fun p => f (SeparationQuotient.mk p.1, SeparationQuotient.mk p.2)
• SeparationQuotient.uniformContinuous_uncurry_lift₂ 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β → γ} (h : ∀ (a : α) (c : β) (b : α) (d : β), Inseparable a b → Inseparable c d → f a c = f b d) : UniformContinuous (Function.uncurry (SeparationQuotient.lift₂ f h)) ↔ UniformContinuous (Function.uncurry f)
• IsUniformInducing.uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : IsUniformInducing f) : UniformContinuous f
• IsUniformInducing.uniformContinuous_iff 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hg : IsUniformInducing g) : UniformContinuous f ↔ UniformContinuous (g ∘ f)
• IsUniformEmbedding.of_comp 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) (hgf : IsUniformEmbedding (g ∘ f)) : IsUniformEmbedding f
• IsUniformInducing.of_comp 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) (hgf : IsUniformInducing (g ∘ f)) : IsUniformInducing f
• UniformContinuous.rangeFactorization 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformContinuous f) : UniformContinuous (Set.rangeFactorization f)
• uniformContinuous_rangeFactorization_iff 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} : UniformContinuous (Set.rangeFactorization f) ↔ UniformContinuous f
• Dense.uniformContinuous_extend 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : ↑s → β} [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) : UniformContinuous (hs.extend f)
• uniformly_extend_exists 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α) : ∃ c, Filter.Tendsto f (Filter.comap e (nhds a)) (nhds c)
• isUniformInducing_iff' 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} : IsUniformInducing f ↔ UniformContinuous f ∧ Filter.comap (Prod.map f f) (uniformity β) ≤ uniformity α
• uniformContinuous_uniformly_extend 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] : UniformContinuous (⋯.extend f)
• isUniformEmbedding_iff' 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} : IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ Filter.comap (Prod.map f f) (uniformity β) ≤ uniformity α
• uniformly_extend_of_ind 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [T0Space γ] (b : β) : ⋯.extend f (e b) = f b
• Dense.extend_of_ind 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : ↑s → β} [T0Space β] (hs : Dense s) (hf : UniformContinuous f) (x : ↑s) : hs.extend f ↑x = f x
• Dense.extend_exists 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : ↑s → β} [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) : ∃ b, Filter.Tendsto f (Filter.comap Subtype.val (nhds a)) (nhds b)
• Equiv.isUniformEmbedding 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] (f : α ≃ β) (h₁ : UniformContinuous ⇑f) (h₂ : UniformContinuous ⇑f.symm) : IsUniformEmbedding ⇑f
• uniformly_extend_spec 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α) : Filter.Tendsto f (Filter.comap e (nhds a)) (nhds (⋯.extend f a))
• Dense.extend_spec 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : ↑s → β} [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) : Filter.Tendsto f (Filter.comap Subtype.val (nhds a)) (nhds (hs.extend f a))
• Filter.HasBasis.isUniformEmbedding_iff 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {ι : Sort u_1} {ι' : Sort u_2} {p : ι → Prop} {p' : ι' → Prop} {s : ι → Set (α × α)} {s' : ι' → Set (β × β)} (h : (uniformity α).HasBasis p s) (h' : (uniformity β).HasBasis p' s') {f : α → β} : IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j
• uniform_extend_subtype 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β} {s : Set α} (hf : UniformContinuous fun x => f ↑x) (he : IsUniformEmbedding e) (hd : ∀ (x : β), x ∈ closure (Set.range e)) (hb : closure (e '' s) ∈ nhds b) (hs : IsClosed s) (hp : ∀ x ∈ s, p x) : ∃ c, Filter.Tendsto f (Filter.comap e (nhds b)) (nhds c)
• Pi.uniformContinuous_proj 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)] (i : ι) : UniformContinuous fun a => a i
• Pi.uniformContinuous_precomp 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} {ι' : Type u_2} {β : Type u_3} [UniformSpace β] (φ : ι' → ι) : UniformContinuous fun x => x ∘ φ
• uniformContinuous_pi 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} {α : ι → Type u} [U : (i : ι) → UniformSpace (α i)] {β : Type u_4} [UniformSpace β] {f : β → (i : ι) → α i} : UniformContinuous f ↔ ∀ (i : ι), UniformContinuous fun x => f x i
• Pi.uniformContinuous_precomp' 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} {ι' : Type u_2} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)] (φ : ι' → ι) : UniformContinuous fun f j => f (φ j)
• Pi.uniformContinuous_postcomp 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} {β : Type u_3} [UniformSpace β] {α : Type u_4} [UniformSpace α] {g : α → β} (hg : UniformContinuous g) : UniformContinuous fun x => g ∘ x
• Pi.uniformContinuous_postcomp' 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)] {β : ι → Type u_4} [(i : ι) → UniformSpace (β i)] {g : (i : ι) → α i → β i} (hg : ∀ (i : ι), UniformContinuous (g i)) : UniformContinuous fun f i => g i (f i)
• Pi.uniformContinuous_restrict 📋 Mathlib.Topology.UniformSpace.Pi
  {ι : Type u_1} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)] (S : Set ι) : UniformContinuous S.domRestrict
• UniformEquiv.uniformContinuous_invFun 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (self : α ≃ᵤ β) : UniformContinuous self.invFun
• UniformEquiv.uniformContinuous_toFun 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (self : α ≃ᵤ β) : UniformContinuous self.toFun
• UniformEquiv.mk 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (toEquiv : α ≃ β) (uniformContinuous_toFun : UniformContinuous toEquiv.toFun) (uniformContinuous_invFun : UniformContinuous toEquiv.invFun) : α ≃ᵤ β
• UniformEquiv.uniformContinuous 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : UniformContinuous ⇑h
• UniformEquiv.uniformContinuous_symm 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : UniformContinuous ⇑h.symm
• UniformEquiv.uniformEquiv_mk_coe 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c } = ⇑a
• UniformEquiv.uniformEquiv_mk_coe_symm 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c }.symm = ⇑a.symm
• UniformContinuous.comp_tendstoLocallyUniformly 📋 Mathlib.Topology.UniformSpace.LocallyUniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (hf : TendstoLocallyUniformly F f p) : TendstoLocallyUniformly (fun x => g ∘ F x) (g ∘ f) p
• UniformContinuous.comp_tendstoLocallyUniformlyOn 📋 Mathlib.Topology.UniformSpace.LocallyUniformConvergence
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (hf : TendstoLocallyUniformlyOn F f p s) : TendstoLocallyUniformlyOn (fun x => g ∘ F x) (g ∘ f) p s
• TendstoUniformly.uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformApproximation
  {α : Type u_4} {β : Type u_5} {ι : Type u_6} [UniformSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} (h : TendstoUniformly F f p) (hc : ∃ᶠ (n : ι) in p, UniformContinuous (F n)) : UniformContinuous f
• uniformContinuous_of_uniform_approx_of_uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformApproximation
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] {f : α → β} (h : ∀ u ∈ uniformity β, ∃ F, UniformContinuous F ∧ ∀ (y : α), (f y, F y) ∈ u) : UniformContinuous f
• UniformFun.uniformContinuous_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} [UniformSpace β] : UniformContinuous ⇑UniformFun.toFun
• UniformFun.uniformContinuous_eval 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (x : α) : UniformContinuous (Function.eval x ∘ ⇑UniformFun.toFun)
• UniformOnFun.uniformContinuous_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} (h : ⋃₀ 𝔖 = Set.univ) : UniformContinuous ⇑(UniformOnFun.toFun 𝔖)
• UniformOnFun.uniformContinuous_eval 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} (h : ⋃₀ 𝔖 = Set.univ) (x : α) : UniformContinuous (Function.eval x ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_eval_of_mem_sUnion 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 : Set (Set α)) {x : α} (hx : x ∈ ⋃₀ 𝔖) : UniformContinuous (Function.eval x ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_eval_of_mem 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) {s : Set α} [UniformSpace β] (𝔖 : Set (Set α)) {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) : UniformContinuous (Function.eval x ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformFun.precomp_uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace β] {f : γ → α} : UniformContinuous fun g => UniformFun.ofFun (UniformFun.toFun g ∘ f)
• UniformOnFun.uniformContinuous_restrict_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} : UniformContinuous ((⋃₀ 𝔖).domRestrict ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_ofUniformFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 : Set (Set α)) : UniformContinuous fun f => (UniformOnFun.ofFun 𝔖) (UniformFun.toFun f)
• UniformFun.postcomp_uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace β] [UniformSpace γ] {f : γ → β} (hf : UniformContinuous f) : UniformContinuous (⇑UniformFun.ofFun ∘ (fun x => f ∘ x) ∘ ⇑UniformFun.toFun)
• UniformOnFun.uniformContinuous_ofFun_toFun_of_subset 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 𝔗 : Set (Set α)) (h : 𝔖 ⊆ 𝔗) : UniformContinuous (⇑(UniformOnFun.ofFun 𝔗) ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.precomp_uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace β] {𝔖 : Set (Set α)} {𝔗 : Set (Set γ)} {f : γ → α} (hf : Set.MapsTo (fun x => f '' x) 𝔗 𝔖) : UniformContinuous fun g => (UniformOnFun.ofFun 𝔗) ((UniformOnFun.toFun 𝔖) g ∘ f)
• UniformOnFun.postcomp_uniformContinuous 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace β] {𝔖 : Set (Set α)} [UniformSpace γ] {f : γ → β} (hf : UniformContinuous f) : UniformContinuous (⇑(UniformOnFun.ofFun 𝔖) ∘ (fun x => f ∘ x) ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_ofFun_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 𝔗 : Set (Set α)) (h : ∀ s ∈ 𝔖, ∃ T ⊆ 𝔗, T.Finite ∧ s ⊆ ⋃₀ T) : UniformContinuous (⇑(UniformOnFun.ofFun 𝔗) ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_restrict 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  (α : Type u_1) (β : Type u_2) {s : Set α} [UniformSpace β] (𝔖 : Set (Set α)) (h : s ∈ 𝔖) : UniformContinuous (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(UniformOnFun.toFun 𝔖))
• UniformOnFun.uniformContinuous_ofFun_toFun_of_mem 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
  {α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 : Set (Set α)) (s : Set α) (h : s ∈ 𝔖) : UniformContinuous (⇑(UniformOnFun.ofFun 𝔖) ∘ ⇑(UniformOnFun.toFun {s}))
• UniformEquicontinuous.uniformContinuous 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} (h : UniformEquicontinuous F) (i : ι) : UniformContinuous (F i)
• uniformEquicontinuous_finite 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] [Finite ι] {F : ι → β → α} : UniformEquicontinuous F ↔ ∀ (i : ι), UniformContinuous (F i)
• uniformEquicontinuous_unique 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] [Unique ι] {F : ι → β → α} : UniformEquicontinuous F ↔ UniformContinuous (F default)
• Set.UniformEquicontinuous.uniformContinuous_of_mem 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {H : Set (β → α)} (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f
• Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {l : Filter ι} [l.NeBot] {F : ι → β → α} {f : β → α} (h₁ : Filter.Tendsto F l (nhds f)) (h₂ : UniformEquicontinuous F) : UniformContinuous f
• uniformEquicontinuous_iff_uniformContinuous 📋 Mathlib.Topology.UniformSpace.Equicontinuity
  {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} : UniformEquicontinuous F ↔ UniformContinuous (⇑UniformFun.ofFun ∘ Function.swap F)
• CompactSpace.uniformContinuous_of_continuous 📋 Mathlib.Topology.UniformSpace.HeineCantor
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] [CompactSpace α] {f : α → β} (h : Continuous f) : UniformContinuous f
• HasCompactMulSupport.uniformContinuous_of_continuous 📋 Mathlib.Topology.UniformSpace.HeineCantor
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} [One β] (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f
• HasCompactSupport.uniformContinuous_of_continuous 📋 Mathlib.Topology.UniformSpace.HeineCantor
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} [Zero β] (h1 : HasCompactSupport f) (h2 : Continuous f) : UniformContinuous f
• Continuous.uniformContinuous_of_tendsto_cocompact 📋 Mathlib.Topology.UniformSpace.HeineCantor
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} {x : β} (h_cont : Continuous f) (hx : Filter.Tendsto f (Filter.cocompact α) (nhds x)) : UniformContinuous f
• uniformContinuous_inv 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] : UniformContinuous fun x => x⁻¹
• uniformContinuous_neg 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] : UniformContinuous fun x => -x
• uniformContinuous_div_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) : UniformContinuous fun b => b / a
• uniformContinuous_sub_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) : UniformContinuous fun b => b - a
• uniformContinuous_const_zsmul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (n : ℤ) : UniformContinuous fun x => n • x
• uniformContinuous_zpow_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (n : ℤ) : UniformContinuous fun x => x ^ n
• uniformContinuous_const_nsmul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (n : ℕ) : UniformContinuous fun x => n • x
• uniformContinuous_pow_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (n : ℕ) : UniformContinuous fun x => x ^ n
• uniformContinuous_add_left 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) : UniformContinuous fun b => a + b
• uniformContinuous_add_right 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) : UniformContinuous fun b => b + a
• uniformContinuous_mul_left 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) : UniformContinuous fun b => a * b
• uniformContinuous_mul_right 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) : UniformContinuous fun b => b * a
• UniformContinuous.inv 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) : UniformContinuous fun x => (f x)⁻¹
• UniformContinuous.neg 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) : UniformContinuous fun x => -f x
• UniformContinuous.div_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => f x / a
• UniformContinuous.sub_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => f x - a
• uniformContinuous_div 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] : UniformContinuous fun p => p.1 / p.2
• uniformContinuous_sub 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] : UniformContinuous fun p => p.1 - p.2
• IsUniformAddGroup.mk 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} [UniformSpace α] [AddGroup α] (uniformContinuous_sub : UniformContinuous fun p => p.1 - p.2) : IsUniformAddGroup α
• IsUniformAddGroup.uniformContinuous_sub 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} {inst✝ : UniformSpace α} {inst✝¹ : AddGroup α} [self : IsUniformAddGroup α] : UniformContinuous fun p => p.1 - p.2
• IsUniformGroup.mk 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} [UniformSpace α] [Group α] (uniformContinuous_div : UniformContinuous fun p => p.1 / p.2) : IsUniformGroup α
• IsUniformGroup.uniformContinuous_div 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} {inst✝ : UniformSpace α} {inst✝¹ : Group α} [self : IsUniformGroup α] : UniformContinuous fun p => p.1 / p.2
• UniformContinuous.const_zsmul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℤ) : UniformContinuous fun x => n • f x
• UniformContinuous.zpow_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℤ) : UniformContinuous fun x => f x ^ n
• UniformContinuous.const_nsmul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℕ) : UniformContinuous fun x => n • f x
• UniformContinuous.pow_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (n : ℕ) : UniformContinuous fun x => f x ^ n
• UniformContinuous.add_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => f x + a
• UniformContinuous.const_add 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => a + f x
• UniformContinuous.const_mul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => a * f x
• UniformContinuous.mul_const 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f : β → α} (hf : UniformContinuous f) (a : α) : UniformContinuous fun x => f x * a
• uniformContinuous_add 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] : UniformContinuous fun p => p.1 + p.2
• uniformContinuous_mul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] : UniformContinuous fun p => p.1 * p.2
• UniformContinuous.div 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun x => f x / g x
• UniformContinuous.sub 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun x => f x - g x
• UniformContinuous.add 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun x => f x + g x
• UniformContinuous.mul 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun x => f x * g x
• Finset.uniformContinuous_prod 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} {β : Type u_4} {ι : Type u_5} [UniformSpace α] [CommGroup α] [IsUniformGroup α] [UniformSpace β] {f : ι → β → α} (s : Finset ι) (h : ∀ i ∈ s, UniformContinuous (f i)) : UniformContinuous fun x => ∏ i ∈ s, f i x
• Finset.uniformContinuous_sum 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} {β : Type u_4} {ι : Type u_5} [UniformSpace α] [AddCommGroup α] [IsUniformAddGroup α] [UniformSpace β] {f : ι → β → α} (s : Finset ι) (h : ∀ i ∈ s, UniformContinuous (f i)) : UniformContinuous fun x => ∑ i ∈ s, f i x
• IsUniformAddGroup.mk' 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} [UniformSpace α] [AddGroup α] (h₁ : UniformContinuous fun p => p.1 + p.2) (h₂ : UniformContinuous fun p => -p) : IsUniformAddGroup α
• IsUniformGroup.mk' 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_3} [UniformSpace α] [Group α] (h₁ : UniformContinuous fun p => p.1 * p.2) (h₂ : UniformContinuous fun p => p⁻¹) : IsUniformGroup α
• uniformContinuous_addMonoidHom_of_continuous 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom} (h : Continuous ⇑f) : UniformContinuous ⇑f
• uniformContinuous_monoidHom_of_continuous 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Continuous ⇑f) : UniformContinuous ⇑f
• uniformContinuous_of_continuousAt_one 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 1) : UniformContinuous ⇑f
• uniformContinuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 0) : UniformContinuous ⇑f
• uniformContinuous_of_tendsto_one 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Filter.Tendsto (⇑f) (nhds 1) (nhds 1)) : UniformContinuous ⇑f
• uniformContinuous_of_tendsto_zero 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom} (h : Filter.Tendsto (⇑f) (nhds 0) (nhds 0)) : UniformContinuous ⇑f
• IsUniformAddGroup.uniformContinuous_iff_isOpen_ker 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {hom : Type u_3} [UniformSpace β] [DiscreteTopology β] [AddGroup β] [IsUniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom} : UniformContinuous ⇑f ↔ IsOpen ↑(↑f).ker
• IsUniformGroup.uniformContinuous_iff_isOpen_ker 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {hom : Type u_3} [UniformSpace β] [DiscreteTopology β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} : UniformContinuous ⇑f ↔ IsOpen ↑(↑f).ker
• AddMonoidHom.uniformContinuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] (f : α →+ β) (hf : ContinuousAt (⇑f) 0) : UniformContinuous ⇑f
• MonoidHom.uniformContinuous_of_continuousAt_one 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
  {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] [Group β] [IsUniformGroup β] (f : α →* β) (hf : ContinuousAt (⇑f) 1) : UniformContinuous ⇑f
• ContinuousLinearMap.uniformContinuous 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {E₁ : Type u_9} {E₂ : Type u_10} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [IsUniformAddGroup E₁] [IsUniformAddGroup E₂] (f : E₁ →SL[σ₁₂] E₂) : UniformContinuous ⇑f
• EMetric.uniformContinuous_iff_le 📋 Mathlib.Topology.EMetricSpace.Defs
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, edist a b ≤ δ → edist (f a) (f b) ≤ ε
• EMetric.uniformContinuous_iff 📋 Mathlib.Topology.EMetricSpace.Defs
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε
• Metric.uniformContinuous_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, dist a b < δ → dist (f a) (f b) < ε
• Metric.uniformContinuous_iff_le 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, dist a b ≤ δ → dist (f a) (f b) ≤ ε
• EMetric.isUniformInducing_iff 📋 Mathlib.Topology.EMetricSpace.Basic
  {γ : Type u} {β : Type v} [PseudoEMetricSpace γ] [PseudoEMetricSpace β] {f : γ → β} : IsUniformInducing f ↔ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ
• EMetric.isUniformEmbedding_iff 📋 Mathlib.Topology.EMetricSpace.Basic
  {γ : Type u} {β : Type v} [PseudoEMetricSpace γ] [PseudoEMetricSpace β] {f : γ → β} : IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ
• Metric.isUniformInducing_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : IsUniformInducing f ↔ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ

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