Theorems · Theorem · general topology

Filter.HasBasis.isUniformInducing_iff

∀ {α : Type u} {β : Type v} [inst : UniformSpace α] [inst_1 : UniformSpace β] {ι : Sort u_1} {ι' : Sort u_2}
  {p : ι → Prop} {p' : ι' → Prop} {s : ι → Set (α × α)} {s' : ι' → Set (β × β)},
  (uniformity α).HasBasis p s →
    (uniformity β).HasBasis p' s' →
      ∀ {f : α → β},
        IsUniformInducing f ↔
          (∀ (i : ι'), p' i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
            ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j

Defined in

Mathlib.Topology.UniformSpace.UniformEmbedding

Cited by

2 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpaceUniformSpace