{α : Type u_1} →
  {β : Type u_2} →
    [inst : TopologicalSpace α] →
      [inst_1 : TopologicalSpace β] →
        {ι : Type u_5} →
          (S : ι → Set α) →
            (φ : (i : ι) → C(↑(S i), β)) →
              (∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩) →
                (∀ (x : α), ∃ i, S i ∈ nhds x) → C(α, β)