Borel hierarchy and omega context free languages

We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:”A Decade of Concurrency”, Springer LNCS 803 (1994), 583-621], proving topological properties of omega context free languages (ω-CFL) which extend those of [O. Finkel, Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Vol. 262 (1-2), 2001, p. 669-697]: there exist some ω-CFL which are non Borel sets and one cannot decide whether an ω-CFL is a Borel set. We give also an answer to a question of Niwinski [Problem on ω-Powers Posed in the Proceedings of the 1990 Workshop ”Logics and Recognizable Sets”] and of Simonnet [Automates et Théorie Descriptive, Ph.D. Thesis, Université Paris 7, March 1992] about ω-powers of finitary languages, giving an example of a finitary context free language L such that L is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an ω-CFL is an arithmetical set. Finally we extend some results to context free sets of infinite trees.