Uniform structure on topological groups #

This file defines uniform groups and its additive counterpart. These typeclasses should be preferred over using [TopologicalSpace α] [IsTopologicalGroup α] since every topological group naturally induces a uniform structure.

Main declarations #

IsUniformGroup and IsUniformAddGroup: Multiplicative and additive uniform groups, i.e., groups with uniformly continuous (*) and (⁻¹) / (+) and (-).

Main results #

IsTopologicalAddGroup.toUniformSpace and comm_topologicalAddGroup_is_uniform can be used to construct a canonical uniformity for a topological add group.

See Mathlib.Topology.Algebra.IsUniformGroup.Basic for further results.

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class IsUniformGroup (α : Type u_3) [UniformSpace α] [Group α] :

Prop

A uniform group is a group in which multiplication and inversion are uniformly continuous.

uniformContinuous_div : UniformContinuous fun (p : α × α) => p.1 / p.2

Instances

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@[deprecated IsUniformGroup (since := "2025-03-26")]

def UniformGroup (α : Type u_3) [UniformSpace α] [Group α] :

Prop

Alias of IsUniformGroup.

A uniform group is a group in which multiplication and inversion are uniformly continuous.

Equations

UniformGroup = IsUniformGroup

Instances For

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class IsUniformAddGroup (α : Type u_3) [UniformSpace α] [AddGroup α] :

Prop

A uniform additive group is an additive group in which addition and negation are uniformly continuous.

uniformContinuous_sub : UniformContinuous fun (p : α × α) => p.1 - p.2

Instances

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@[deprecated IsUniformAddGroup (since := "2025-03-26")]

def UniformAddGroup (α : Type u_3) [UniformSpace α] [AddGroup α] :

Prop

Alias of IsUniformAddGroup.

A uniform additive group is an additive group in which addition and negation are uniformly continuous.

Equations

UniformAddGroup = IsUniformAddGroup

Instances For

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theorem IsUniformGroup.mk' {α : Type u_3} [UniformSpace α] [Group α] (h₁ : UniformContinuous fun (p : α × α) => p.1 * p.2) (h₂ : UniformContinuous fun (p : α) => p⁻¹) :

IsUniformGroup α

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theorem IsUniformAddGroup.mk' {α : Type u_3} [UniformSpace α] [AddGroup α] (h₁ : UniformContinuous fun (p : α × α) => p.1 + p.2) (h₂ : UniformContinuous fun (p : α) => -p) :

IsUniformAddGroup α

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theorem uniformContinuous_div {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

UniformContinuous fun (p : α × α) => p.1 / p.2

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theorem uniformContinuous_sub {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

UniformContinuous fun (p : α × α) => p.1 - p.2

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theorem UniformContinuous.div {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 : β) => f x / g x

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theorem UniformContinuous.sub {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 : β) => f x - g x

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theorem UniformContinuous.inv {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 : β) => (f x)⁻¹

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theorem UniformContinuous.neg {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 : β) => -f x

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theorem uniformContinuous_inv {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

UniformContinuous fun (x : α) => x⁻¹

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theorem uniformContinuous_neg {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

UniformContinuous fun (x : α) => -x

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theorem UniformContinuous.mul {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 : β) => f x * g x

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theorem UniformContinuous.add {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 : β) => f x + g x

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theorem uniformContinuous_mul {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

UniformContinuous fun (p : α × α) => p.1 * p.2

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theorem uniformContinuous_add {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

UniformContinuous fun (p : α × α) => p.1 + p.2

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theorem UniformContinuous.mul_const {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 : β) => f x * a

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theorem UniformContinuous.add_const {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 : β) => f x + a

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theorem UniformContinuous.const_mul {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 : β) => a * f x

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theorem UniformContinuous.const_add {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 : β) => a + f x

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theorem uniformContinuous_mul_left {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) :

UniformContinuous fun (b : α) => a * b

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theorem uniformContinuous_add_left {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) :

UniformContinuous fun (b : α) => a + b

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theorem uniformContinuous_mul_right {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) :

UniformContinuous fun (b : α) => b * a

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theorem uniformContinuous_add_right {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) :

UniformContinuous fun (b : α) => b + a

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theorem UniformContinuous.div_const {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 : β) => f x / a

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theorem UniformContinuous.sub_const {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 : β) => f x - a

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theorem uniformContinuous_div_const {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) :

UniformContinuous fun (b : α) => b / a

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theorem uniformContinuous_sub_const {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) :

UniformContinuous fun (b : α) => b - a

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theorem UniformContinuous.pow_const {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 : β) => f x ^ n

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theorem UniformContinuous.const_nsmul {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 : β) => n • f x

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theorem uniformContinuous_pow_const {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (n : ℕ) :

UniformContinuous fun (x : α) => x ^ n

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theorem uniformContinuous_const_nsmul {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (n : ℕ) :

UniformContinuous fun (x : α) => n • x

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theorem UniformContinuous.zpow_const {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 : β) => f x ^ n

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theorem UniformContinuous.const_zsmul {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 : β) => n • f x

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theorem uniformContinuous_zpow_const {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (n : ℤ) :

UniformContinuous fun (x : α) => x ^ n

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theorem uniformContinuous_const_zsmul {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (n : ℤ) :

UniformContinuous fun (x : α) => n • x

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@[instance 10]

instance IsUniformGroup.to_topologicalGroup {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

IsTopologicalGroup α

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@[instance 10]

instance IsUniformAddGroup.to_topologicalAddGroup {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

IsTopologicalAddGroup α

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@[deprecated IsUniformAddGroup.to_topologicalAddGroup (since := "2025-03-31")]

theorem UniformGroup.to_topologicalAddGroup {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

IsTopologicalAddGroup α

Alias of IsUniformAddGroup.to_topologicalAddGroup.

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@[deprecated IsUniformGroup.to_topologicalGroup (since := "2025-03-31")]

theorem UniformGroup.to_topologicalGroup {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

IsTopologicalGroup α

Alias of IsUniformGroup.to_topologicalGroup.

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instance Prod.instIsUniformGroup {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] [Group β] [IsUniformGroup β] :

IsUniformGroup (α × β)

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instance Prod.instIsUniformAddGroup {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] :

IsUniformAddGroup (α × β)

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@[deprecated Prod.instIsUniformAddGroup (since := "2025-03-31")]

theorem Prod.instUniformAddGroup {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] :

IsUniformAddGroup (α × β)

Alias of Prod.instIsUniformAddGroup.

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@[deprecated Prod.instIsUniformGroup (since := "2025-03-31")]

theorem Prod.instUniformGroup {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] [Group β] [IsUniformGroup β] :

IsUniformGroup (α × β)

Alias of Prod.instIsUniformGroup.

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theorem uniformity_translate_mul {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] (a : α) :

Filter.map (fun (x : α × α) => (x.1 * a, x.2 * a)) (uniformity α) = uniformity α

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theorem uniformity_translate_add {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] (a : α) :

Filter.map (fun (x : α × α) => (x.1 + a, x.2 + a)) (uniformity α) = uniformity α

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instance MulOpposite.instIsUniformGroup {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

IsUniformGroup αᵐᵒᵖ

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instance AddOpposite.instIsUniformAddGroup {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

IsUniformAddGroup αᵃᵒᵖ

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theorem isUniformGroup_sInf {β : Type u_2} [Group β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, IsUniformGroup β) :

IsUniformGroup β

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theorem isUniformAddGroup_sInf {β : Type u_2} [AddGroup β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, IsUniformAddGroup β) :

IsUniformAddGroup β

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@[deprecated isUniformAddGroup_sInf (since := "2025-03-31")]

theorem uniformAddGroup_sInf {β : Type u_2} [AddGroup β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, IsUniformAddGroup β) :

IsUniformAddGroup β

Alias of isUniformAddGroup_sInf.

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@[deprecated isUniformGroup_sInf (since := "2025-03-31")]

theorem uniformGroup_sInf {β : Type u_2} [Group β] {us : Set (UniformSpace β)} (h : ∀ u ∈ us, IsUniformGroup β) :

IsUniformGroup β

Alias of isUniformGroup_sInf.

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theorem isUniformGroup_iInf {β : Type u_2} [Group β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), IsUniformGroup β) :

IsUniformGroup β

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theorem isUniformAddGroup_iInf {β : Type u_2} [AddGroup β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), IsUniformAddGroup β) :

IsUniformAddGroup β

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@[deprecated isUniformAddGroup_iInf (since := "2025-03-31")]

theorem uniformAddGroup_iInf {β : Type u_2} [AddGroup β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), IsUniformAddGroup β) :

IsUniformAddGroup β

Alias of isUniformAddGroup_iInf.

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@[deprecated isUniformGroup_iInf (since := "2025-03-31")]

theorem uniformGroup_iInf {β : Type u_2} [Group β] {ι : Sort u_3} {us' : ι → UniformSpace β} (h' : ∀ (i : ι), IsUniformGroup β) :

IsUniformGroup β

Alias of isUniformGroup_iInf.

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theorem isUniformGroup_inf {β : Type u_2} [Group β] {u₁ u₂ : UniformSpace β} (h₁ : IsUniformGroup β) (h₂ : IsUniformGroup β) :

IsUniformGroup β

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theorem isUniformAddGroup_inf {β : Type u_2} [AddGroup β] {u₁ u₂ : UniformSpace β} (h₁ : IsUniformAddGroup β) (h₂ : IsUniformAddGroup β) :

IsUniformAddGroup β

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@[deprecated isUniformAddGroup_inf (since := "2025-03-31")]

theorem uniformAddGroup_inf {β : Type u_2} [AddGroup β] {u₁ u₂ : UniformSpace β} (h₁ : IsUniformAddGroup β) (h₂ : IsUniformAddGroup β) :

IsUniformAddGroup β

Alias of isUniformAddGroup_inf.

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@[deprecated isUniformGroup_inf (since := "2025-03-31")]

theorem uniformGroup_inf {β : Type u_2} [Group β] {u₁ u₂ : UniformSpace β} (h₁ : IsUniformGroup β) (h₂ : IsUniformGroup β) :

IsUniformGroup β

Alias of isUniformGroup_inf.

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theorem uniformity_eq_comap_nhds_one (α : Type u_1) [UniformSpace α] [Group α] [IsUniformGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.2 / x.1) (nhds 1)

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theorem uniformity_eq_comap_nhds_zero (α : Type u_1) [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.2 - x.1) (nhds 0)

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theorem uniformity_eq_comap_nhds_one_swapped (α : Type u_1) [UniformSpace α] [Group α] [IsUniformGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.1 / x.2) (nhds 1)

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theorem uniformity_eq_comap_nhds_zero_swapped (α : Type u_1) [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.1 - x.2) (nhds 0)

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theorem IsUniformGroup.ext {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : IsUniformGroup G) (hv : IsUniformGroup G) (h : nhds 1 = nhds 1) :

u = v

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theorem IsUniformAddGroup.ext {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : IsUniformAddGroup G) (hv : IsUniformAddGroup G) (h : nhds 0 = nhds 0) :

u = v

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@[deprecated IsUniformAddGroup.ext (since := "2025-03-31")]

theorem UniformAddGroup.ext {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : IsUniformAddGroup G) (hv : IsUniformAddGroup G) (h : nhds 0 = nhds 0) :

u = v

Alias of IsUniformAddGroup.ext.

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@[deprecated IsUniformGroup.ext (since := "2025-03-31")]

theorem UniformGroup.ext {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : IsUniformGroup G) (hv : IsUniformGroup G) (h : nhds 1 = nhds 1) :

u = v

Alias of IsUniformGroup.ext.

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theorem IsUniformGroup.ext_iff {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : IsUniformGroup G) (hv : IsUniformGroup G) :

u = v ↔ nhds 1 = nhds 1

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theorem IsUniformAddGroup.ext_iff {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : IsUniformAddGroup G) (hv : IsUniformAddGroup G) :

u = v ↔ nhds 0 = nhds 0

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@[deprecated IsUniformAddGroup.ext_iff (since := "2025-03-31")]

theorem UniformAddGroup.ext_iff {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : IsUniformAddGroup G) (hv : IsUniformAddGroup G) :

u = v ↔ nhds 0 = nhds 0

Alias of IsUniformAddGroup.ext_iff.

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@[deprecated IsUniformGroup.ext_iff (since := "2025-03-31")]

theorem UniformGroup.ext_iff {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : IsUniformGroup G) (hv : IsUniformGroup G) :

u = v ↔ nhds 1 = nhds 1

Alias of IsUniformGroup.ext_iff.

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theorem IsUniformGroup.uniformity_countably_generated {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] [(nhds 1).IsCountablyGenerated] :

(uniformity α).IsCountablyGenerated

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theorem IsUniformAddGroup.uniformity_countably_generated {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [(nhds 0).IsCountablyGenerated] :

(uniformity α).IsCountablyGenerated

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@[deprecated IsUniformAddGroup.uniformity_countably_generated (since := "2025-03-31")]

theorem UniformAddGroup.uniformity_countably_generated {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [(nhds 0).IsCountablyGenerated] :

(uniformity α).IsCountablyGenerated

Alias of IsUniformAddGroup.uniformity_countably_generated.

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@[deprecated IsUniformGroup.uniformity_countably_generated (since := "2025-03-31")]

theorem UniformGroup.uniformity_countably_generated {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] [(nhds 1).IsCountablyGenerated] :

(uniformity α).IsCountablyGenerated

Alias of IsUniformGroup.uniformity_countably_generated.

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theorem uniformity_eq_comap_inv_mul_nhds_one {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.1⁻¹ * x.2) (nhds 1)

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theorem uniformity_eq_comap_neg_add_nhds_zero {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => -x.1 + x.2) (nhds 0)

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theorem uniformity_eq_comap_inv_mul_nhds_one_swapped {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => x.2⁻¹ * x.1) (nhds 1)

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theorem uniformity_eq_comap_neg_add_nhds_zero_swapped {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] :

uniformity α = Filter.comap (fun (x : α × α) => -x.2 + x.1) (nhds 0)

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theorem Filter.HasBasis.uniformity_of_nhds_one {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 1).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.2 / x.1 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_zero {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 0).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.2 - x.1 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 1).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.1⁻¹ * x.2 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_zero_neg_add {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 0).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | -x.1 + x.2 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_one_swapped {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 1).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.1 / x.2 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_zero_swapped {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 0).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.1 - x.2 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped {α : Type u_1} [UniformSpace α] [Group α] [IsUniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 1).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | x.2⁻¹ * x.1 ∈ U i}

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theorem Filter.HasBasis.uniformity_of_nhds_zero_neg_add_swapped {α : Type u_1} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (nhds 0).HasBasis p U) :

(uniformity α).HasBasis p fun (i : ι) => {x : α × α | -x.2 + x.1 ∈ U i}

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theorem uniformContinuous_of_tendsto_one {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

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theorem uniformContinuous_of_tendsto_zero {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

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theorem uniformContinuous_of_continuousAt_one {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

A group homomorphism (a bundled morphism of a type that implements MonoidHomClass) between two uniform groups is uniformly continuous provided that it is continuous at one. See also continuous_of_continuousAt_one.

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theorem uniformContinuous_of_continuousAt_zero {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

An additive group homomorphism (a bundled morphism of a type that implements AddMonoidHomClass) between two uniform additive groups is uniformly continuous provided that it is continuous at zero. See also continuous_of_continuousAt_zero.

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theorem MonoidHom.uniformContinuous_of_continuousAt_one {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

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theorem AddMonoidHom.uniformContinuous_of_continuousAt_zero {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

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theorem IsUniformGroup.uniformContinuous_iff_isOpen_ker {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

A homomorphism from a uniform group to a discrete uniform group is continuous if and only if its kernel is open.

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theorem IsUniformAddGroup.uniformContinuous_iff_isOpen_ker {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

A homomorphism from a uniform additive group to a discrete uniform additive group is continuous if and only if its kernel is open.

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@[deprecated IsUniformAddGroup.uniformContinuous_iff_isOpen_ker (since := "2024-11-18")]

theorem UniformAddGroup.uniformContinuous_iff_open_ker {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

Alias of IsUniformAddGroup.uniformContinuous_iff_isOpen_ker.

A homomorphism from a uniform additive group to a discrete uniform additive group is continuous if and only if its kernel is open.

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@[deprecated IsUniformGroup.uniformContinuous_iff_isOpen_ker (since := "2024-11-18")]

theorem UniformGroup.uniformContinuous_iff_open_ker {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

Alias of IsUniformGroup.uniformContinuous_iff_isOpen_ker.

A homomorphism from a uniform group to a discrete uniform group is continuous if and only if its kernel is open.

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@[deprecated IsUniformAddGroup.uniformContinuous_iff_isOpen_ker (since := "2025-03-30")]

theorem UniformAddGroup.uniformContinuous_iff_isOpen_ker {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

Alias of IsUniformAddGroup.uniformContinuous_iff_isOpen_ker.

A homomorphism from a uniform additive group to a discrete uniform additive group is continuous if and only if its kernel is open.

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@[deprecated IsUniformGroup.uniformContinuous_iff_isOpen_ker (since := "2025-03-30")]

theorem UniformGroup.uniformContinuous_iff_isOpen_ker {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

Alias of IsUniformGroup.uniformContinuous_iff_isOpen_ker.

A homomorphism from a uniform group to a discrete uniform group is continuous if and only if its kernel is open.

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theorem uniformContinuous_monoidHom_of_continuous {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

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theorem uniformContinuous_addMonoidHom_of_continuous {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

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def IsTopologicalGroup.toUniformSpace (G : Type u_1) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] :

UniformSpace G

The right uniformity on a topological group (as opposed to the left uniformity).

Warning: in general the right and left uniformities do not coincide and so one does not obtain a IsUniformGroup structure. Two important special cases where they do coincide are for commutative groups (see comm_topologicalGroup_is_uniform) and for compact groups (see topologicalGroup_is_uniform_of_compactSpace).

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IsTopologicalGroup.toUniformSpace G = { toTopologicalSpace := inst✝¹, uniformity := Filter.comap (fun (p : G × G) => p.2 / p.1) (nhds 1), symm := ⋯, comp := ⋯, nhds_eq_comap_uniformity := ⋯ }

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def IsTopologicalAddGroup.toUniformSpace (G : Type u_1) [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] :

UniformSpace G

The right uniformity on a topological additive group (as opposed to the left uniformity).

Warning: in general the right and left uniformities do not coincide and so one does not obtain a IsUniformAddGroup structure. Two important special cases where they do coincide are for commutative additive groups (see comm_topologicalAddGroup_is_uniform) and for compact additive groups (see topologicalAddGroup_is_uniform_of_compactSpace).

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