This paper is a study of topological properties of omega context free languages (omega-CFL). We first extend some decidability results for the deterministic ones (omega-DCFL), proving that one can decide whether an omega-DCFL is in a given Borel class, or in the Wadge class of a given omega regular language. We prove that omega-CFL exhaust the hierarchy of Borel sets of finite rank, and that one cannot decide the borel class of an omega-CFL, giving an answer to a question of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621]. We give also a (partial) answer to a question of Simonnet about omega powers of finitary languages. We show that Büchi-Landweber's Theorem cannot be extended to even closed omega-CFL: in a Gale-Stewart game with a (closed) omega-CFL winning set, one cannot decide which player has a winning strategy. From the proof of topological properties we derive some arithmetical properties of omega-CFL.
Topological Properties of Omega Context Free Languages