On The Borel Complexity Of Mso Definable Sets Of Branches

An infinite binary word can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Sigma(0)(2) over the Cantor discontinuum. Note that the last coincides with the Borel complexity of omega-regular languages.