mathlib documentation

topology.uniform_space.pi

Indexed product of uniform spaces #

@[instance]
def Pi.uniform_space {ι : Type u_1} (α : ι → Type u) [U : Π (i : ι), uniform_space (α i)] :
uniform_space (Π (i : ι), α i)
Equations
theorem Pi.uniformity {ι : Type u_1} (α : ι → Type u) [U : Π (i : ι), uniform_space (α i)] :
𝓤 (Π (i : ι), α i) = ⨅ (i : ι), filter.comap (λ (a : (Π (i : ι), α i) × Π (i : ι), α i), (a.fst i, a.snd i)) (𝓤 (α i))
theorem Pi.uniform_continuous_proj {ι : Type u_1} (α : ι → Type u) [U : Π (i : ι), uniform_space (α i)] (i : ι) :
uniform_continuous (λ (a : Π (i : ι), α i), a i)
@[instance]
def Pi.complete {ι : Type u_1} (α : ι → Type u) [U : Π (i : ι), uniform_space (α i)] [∀ (i : ι), complete_space (α i)] :
complete_space (Π (i : ι), α i)
@[instance]
def Pi.separated {ι : Type u_1} (α : ι → Type u) [U : Π (i : ι), uniform_space (α i)] [∀ (i : ι), separated_space (α i)] :
separated_space (Π (i : ι), α i)