On Infinite Real Trace Rational Languages of Maximum Topological Complexity

We consider the set Rω(Γ, D) of infinite real traces, over a dependence alphabet (Γ, D) with no isolated letter, equipped with the topology induced by the prefix metric. We then prove that all rational languages of infinite real traces are analytic sets. We reprove also that there exist some rational languages of infinite real traces which are analytic but non Borel sets, and even Σ11-complete, hence of maximum possible topological complexity. For that purpose we give an example of Σ11-complete language which is fundamentally different from the known example of Σ11-complete infinitary rational relation given in [Fin03c].