mathlib documentation

topology.algebra.uniform_group

Uniform structure on topological groups #

@[class]
structure uniform_add_group (α : Type u_3) [uniform_space α] [add_group α] :
Prop

A uniform (additive) group is a group in which the addition and negation are uniformly continuous.

Instances
theorem uniform_add_group.mk' {α : Type u_1} [uniform_space α] [add_group α] (h₁ : uniform_continuous (λ (p : α × α), p.fst + p.snd)) (h₂ : uniform_continuous (λ (p : α), -p)) :
theorem uniform_continuous_sub {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] :
uniform_continuous (λ (p : α × α), p.fst - p.snd)
theorem uniform_continuous.sub {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] {f g : β → α} (hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (λ (x : β), f x - g x)
theorem uniform_continuous.neg {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] {f : β → α} (hf : uniform_continuous f) :
uniform_continuous (λ (x : β), -f x)
theorem uniform_continuous_neg {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] :
uniform_continuous (λ (x : α), -x)
theorem uniform_continuous.add {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] {f g : β → α} (hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (λ (x : β), f x + g x)
theorem uniform_continuous_add {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] :
uniform_continuous (λ (p : α × α), p.fst + p.snd)
@[protected, instance]
def prod.uniform_add_group {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] [add_group β] [uniform_add_group β] :
theorem uniformity_translate {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] (a : α) :
filter.map (λ (x : α × α), (x.fst + a, x.snd + a)) (𝓤 α) = 𝓤 α
theorem uniform_embedding_translate {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] (a : α) :
uniform_embedding (λ (x : α), x + a)
theorem uniformity_eq_comap_nhds_zero (α : Type u_1) [uniform_space α] [add_group α] [uniform_add_group α] :
𝓤 α = filter.comap (λ (x : α × α), x.snd - x.fst) (𝓝 0)
theorem group_separation_rel {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] (x y : α) :
(x, y) 𝓢 α x - y closure {0}
theorem uniform_continuous_of_tendsto_zero {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] [add_group β] [uniform_add_group β] {f : α →+ β} (h : filter.tendsto f (𝓝 0) (𝓝 0)) :
theorem uniform_continuous_of_continuous {α : Type u_1} {β : Type u_2} [uniform_space α] [add_group α] [uniform_add_group α] [uniform_space β] [add_group β] [uniform_add_group β] {f : α →+ β} (h : continuous f) :
theorem cauchy_seq.add {α : Type u_1} [uniform_space α] [add_group α] [uniform_add_group α] {ι : Type u_2} [semilattice_sup ι] {u v : ι → α} (hu : cauchy_seq u) (hv : cauchy_seq v) :

The right uniformity on a topological group.

Equations
theorem topological_add_group.separated_of_zero_sep {G : Type u} [add_comm_group G] [topological_space G] [topological_add_group G] (H : ∀ (x : G), x 0(∃ (U : set G) (H : U 𝓝 0), x U)) :
theorem tendsto_sub_comap_self {α : Type u_1} {β : Type u_2} [topological_space α] [add_comm_group α] [topological_add_group α] [topological_space β] [add_comm_group β] {e : β →+ α} (de : dense_inducing e) (x₀ : α) :
filter.tendsto (λ (t : β × β), t.snd - t.fst) (filter.comap (λ (p : β × β), (e p.fst, e p.snd)) (𝓝 (x₀, x₀))) (𝓝 0)
theorem dense_inducing.extend_Z_bilin {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {G : Type u_5} [topological_space α] [add_comm_group α] [topological_add_group α] [topological_space β] [add_comm_group β] [topological_add_group β] [topological_space γ] [add_comm_group γ] [topological_add_group γ] [topological_space δ] [add_comm_group δ] [topological_add_group δ] [uniform_space G] [add_comm_group G] [uniform_add_group G] [separated_space G] [complete_space G] {e : β →+ α} (de : dense_inducing e) {f : δ →+ γ} (df : dense_inducing f) {φ : β →+ δ →+ G} (hφ : continuous (λ (p : β × δ), (φ p.fst) p.snd)) :
continuous (_.extend (λ (p : β × δ), (φ p.fst) p.snd))

Bourbaki GT III.6.5 Theorem I: ℤ-bilinear continuous maps from dense images into a complete Hausdorff group extend by continuity. Note: Bourbaki assumes that α and β are also complete Hausdorff, but this is not necessary.