There Exist Some omega -Powers of Any Borel Rank

The operation V → V ω is a fundamental operation over finitary languages leading to ω-languages. Since the set Σ of infinite words over a finite alphabet Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers of finitary languages naturally arises and has been posed by Niwinski [Niw90], Simonnet [Sim92] and Staiger [Sta97a]. It has been recently proved that for each integer n ≥ 1, there exist some ω-powers of context free languages which are Π0n-complete Borel sets, [Fin01], that there exists a context free language L such that L is analytic but not Borel, [Fin03], and that there exists a finitary language V such that V ω is a Borel set of infinite rank, [Fin04]. But it was still unknown which could be the possible infinite Borel ranks of ω-powers. We fill this gap here, proving the following very surprising result which shows that ω-powers exhibit a great topological complexity: for each non-null countable ordinal ξ, there exist some Σ0ξ-complete ω-powers, and some Π 0 ξ-complete ω-powers.