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

Finkel O.

Fundamenta Informaticae

2004

Vol. 62, nr 3,4

333--342

ω-powers of finitary languages are ω-languages s in the form Vω, where V is a finitary language over a finite alphabet Σ. Since the set Σ,sup>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 [4] that for each integer n ≥ 1, there exist some ω-powers of context free languages which are Πn0-complete Borel sets, and in [5] that there exists a context free language L such that Lω is analytic but not Borel. But the question was still open whether there exists a finitary language V such that Vω 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.