∀ {ι : Type u_1} {R : ι → Type u_2} {A : (i : ι) → Set (R i)} [inst : (i : ι) → TopologicalSpace (R i)],
  (∀ (i : ι), IsOpen (A i)) → ∀ {S : Set ι}, Filter.cofinite ≤ Filter.principal S → IsOpen {f | ∀ (i : ι), ↑f i ∈ A i}