  (Think of the case where k + 1 = 3. Here A is a subset of three-dimensional space and p[A] is the result of “projecting” A along the third axis onto the plane spanned by the first two axes.)  We can now define the hierarchy of projective sets as follows: At the base level, let Σ̰10 consist of the open subsets of (ωω)k and let Π̰10 consist of the closed subsets of (ωω)k. For each n such that 0 &lt; n &lt; ω, recursively define Π̰1n to consist of the complements of sets in Σ̰1n, and define Σ̰1n+1 to consist of the projections of sets in Π̰1n. The projective sets of (k-tuples of) reals are the sets appearing in this hierarchy. This hierarchy is also a proper hierarchy, as can be seen using universal sets and diagonalization. 