nhdsWithin_pi_eq' — Mathlib

English

Continuity at a is equivalent to the Tendsto property from the punctured neighborhood to the target neighborhood.

Русский

Непрерывность в точке a эквивалентна свойству стремления функции к образу через punctured окрестности.

LaTeX

$$$$ ContinuousAt(f,a) \\iff Tendsto(f, 𝓝_{\\neq} a, 𝓝(f(a))) $$$$

Lean4

theorem nhdsWithin_pi_eq' {I : Set ι} (hI : I.Finite) (s : ∀ i, Set (X i)) (x : ∀ i, X i) :
    𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by
  simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ← iInf_principal_finite hI,
    ← iInf_inf_eq]

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