 Now assume that "n" ≥ 1 and the sets "U"("k" / 2"n") and "V"("k" / 2"n") have already been constructed for "k" = 1, ..., 2"n"−1.Since "X" is normal, for any "a" ∈ { 0, 1, ..., 2"n"−1 }, we can find two disjoint open sets which contain "X" \ "V"("a" / 2"n") and "X" \ "U"(("a"+1) / 2"n"), respectively.Call these two open sets "U"((2"a"+1) / 2"n"+1) and "V"((2"a"+1) / 2"n"+1), and verify the above three conditions.