  The Borel sets of reals (or sets of k-tuples of reals) are obtained by starting with the closed subsets of ωω (or (ωω)k for some k &lt; ω) and closing under the operations of countable union and countable intersection. This can be described level by level as follows: Let k &lt; ω. At the base level, let Σ̰01 consist of the open subsets of (ωω)k and let Π̰01 consist of the closed subsets of (ωω)k. For each ordinal α such that 0 &lt; α &lt; ω1, recursively define Σ̰0α to consist of the sets that are countable unions of sets appearing in some Π̰0β, for β &lt; α, and define Π̰0α to consist of the sets that are countable intersections of sets appearing in some Σ̰0β, for β &lt; α. The Borel sets of (k-tuples of) reals are the sets appearing in this   hierarchy.[10]   In 1905 Lebesgue employed Cantor's notion of a universal set and technique of diagonalization to show that the hierarchy of Borel sets is a proper hierarchy (that is, new sets appear at each level). 