has_seq_basis — Mathlib

English

The neighborhood of a product point equals the lifted product of neighborhoods, via the uniformity lift.

Русский

Окрестность произведения равна подъемной окрестности через униформность.

LaTeX

$$$\\mathcal{N}(a,b) = (\\mathcal{U} α).lift'\\bigl( s \\mapsto \\{y: α\\mid (y,a) ∈ s\\} \\times \\{y: α\\mid (b,y) ∈ s\\} \\bigr)$$$

Lean4

theorem has_seq_basis [IsCountablyGenerated <| 𝓤 α] :
    ∃ V : ℕ → Set (α × α), HasAntitoneBasis (𝓤 α) V ∧ ∀ n, IsSymmetricRel (V n) :=
  let ⟨U, hsym, hbasis⟩ := (@UniformSpace.hasBasis_symmetric α _).exists_antitone_subbasis
  ⟨U, hbasis, fun n => (hsym n).2⟩

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