Topological properties of omega context-free languages

This paper is a study of topological properties of omega context free languages (ω-CFL). We first extend some decidability results for the deterministic ones (ωDCFL), proving that one can decide whether an ω-DCFL is in a given Borel class, or in the Wadge class of a given ω-regular language . We prove that ω-CFL exhaust the hierarchy of Borel sets of finite rank, and that one cannot decide the borel class of an ω-CFL, giving an answer to a question of [LT94]. We give also a (partial) answer to a question of [Sim92] about omega powers of finitary languages. We show that Büchi-Landweber’s Theorem cannot be extended to even closed ω-CFL: in a Gale-Stewart game with a (closed) ω-CFL winning set, one cannot decide which player has a winning strategy. From the proof of topological properties we derive some arithmetical properties of ω-CFL.