Theorems · Theorem · general topology

UniformFun.hasBasis_nhds_of_basis

∀ (α : Type u_1) (β : Type u_2) {ι : Type u_4} [inst : UniformSpace β] (f : UniformFun α β) {p : ι → Prop}
  {s : ι → Set (β × β)},
  (uniformity β).HasBasis p s → (nhds f).HasBasis p fun i => {g | (f, g) ∈ UniformFun.gen α β (s i)}

For f : α →ᵤ β, 𝓝 f admits the family {g | ∀ x, (f x, g x) ∈ V} for V ∈ 𝓑 as a filter basis, for any basis 𝓑 of 𝓤 β.

Defined in

Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Cited by

7 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpace