On Infinite Real Trace Rational Languages of Maximum Topological   Complexity
  We consider the set of infinite real traces, over a dependence alphabet (Gamma, 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 and that there exist some rational languages of infinite real traces which are analytic but non Borel sets, and even Sigma^1_1-complete, hence of maximum possible topological complexity. 
Olivier Finkel (ELM), Jean-Pierre Ressayre (ELM), Pierre Simonnet   (SPE)