An ω-Power of a Finitary Language Which is a Borel Set of Infinite Rank

ω-powers of finitary languages are ω-languages in the form V^ω, where V is a finitary language over a finite alphabet Sigma. Since the set Σ^ω of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [13], by Simonnet [15], and by Staiger [18]. It has been proved in [14] that for each integer n≥1, there exist some ω-powers of context free languages which are Π_n⁰-complete Borel sets, and in [5] that there exists a context free language L such that L^ω is analytic but not Borel. But the question was still open whether there exists a finitary language V such that V^ω is a Borel set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose ω-power is Borel of infinite rank.