Theorems · Theorem · general topology

Filter.HasBasis.uniformEquicontinuousOn_iff

∀ {ι : Type u_1} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {κ₁ : Type u_11}
  {κ₂ : Type u_12} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
  {S : Set β},
  (uniformity β ⊓ Filter.principal (S ×ˢ S)).HasBasis p₁ s₁ →
    (uniformity α).HasBasis p₂ s₂ →
      (UniformEquicontinuousOn F S ↔
        ∀ (k₂ : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ (x y : β), (x, y) ∈ s₁ k₁ → ∀ (i : ι), (F i x, F i y) ∈ s₂ k₂)

Defined in

Mathlib.Topology.UniformSpace.Equicontinuity

Cited by

0 results in Mathlib

Foundations

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

Assumes

UniformSpaceUniformSpace