preimage_continuous — Mathlib

English

If f is locally finite and g is another family, then the product right construction preserves local finiteness.

Русский

Если f локально конечна и задана правая конструкция произведения над g, локальная конечность сохраняется.

LaTeX

$$(hf : LocallyFinite f) (g : ι → Set Y) → LocallyFinite (fun i ↦ f i ×ˢ g i)$$

Lean4

theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) : LocallyFinite (g ⁻¹' f ·) :=
  fun x =>
  let ⟨s, hsx, hs⟩ := hf (g x)
  ⟨g ⁻¹' s, hg.continuousAt hsx, hs.subset fun _ ⟨y, hy⟩ => ⟨g y, hy⟩⟩

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