An omega-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 , where V is a finitary language over a finite alphabet Σ. Since the set Σ 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 [Niw90], by Simonnet [Sim92], and by Staiger [Sta97b]. It has been proved in [Fin01] that for each integer n ≥ 1, there exist some ω-powers of context free languages which are Πn-complete Borel sets, and in [Fin03] that there exists a context free language L such that L 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.