congr_uniformity — Mathlib

English

If α and β have countably generated uniformities, then 𝓤(α ⊕ β) is countably generated.

Русский

Если 𝓤 α и 𝓤 β счётно порождены, то 𝓤(α ⊕ β) тоже счётно порождение.

LaTeX

$$IsCountablyGenerated(𝓤(α ⊕ β)) given IsCountablyGenerated(𝓤 α) and IsCountablyGenerated(𝓤 β)$$

Lean4

theorem congr_uniformity {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β} (hf : Tendsto f l (𝓝 b))
    (hg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto g l (𝓝 b) :=
  Uniform.tendsto_nhds_right.2 <| (Uniform.tendsto_nhds_right.1 hf).uniformity_trans hg

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