Unambiguous Tree Languages Are Topologically Harder Than Deterministic Ones

The paper gives an example of a tree language G that is recognised by an unambiguous parity automaton and is Σ1-complete (analytic-complete) as a set in Cantor space. This already shows that the unambiguous languages are topologically more complex than the deterministic ones, that are all in Π1. Using set G as a building block we construct an unambiguous language that is topologically harder than any countable boolean combination of Σ1 and Π 1 1 sets. In particular the language is harder than any set in difference hierarchy of analytic sets considered by O. Finkel and P. Simonnet in the context of nondeterministic automata.