English
A variant of the previous equivalence tailored to an abstract setting for pi spaces.
Русский
Вариант предыдущего эквивалентного условия для абстрактного окружения Pi-пространств.
LaTeX
$$$\\displaystyle \\text{set\_pi\_mem\_nhds\_iff'} \\approx \\text{(equivalent formulation for finite pi-neighborhoods)}$$$
Lean4
theorem set_pi_mem_nhds_iff {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (A i)} (a : ∀ i, A i) :
I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) := by rw [nhds_pi, pi_mem_pi_iff hI]