This talk will be a continuation of my talk from 11th February. I will not assume knowledge of any material presented then except for basic notions of topology. During this talk I will move to the case of regular languages of infinite trees. Since there are known examples of such languages that are not Borel I will introduce the projective hierarchy. This hierarchy allows to study complexity of sets defined using big quantifiers (i.e. ranging over sets of naturals). I will focus on correspondence between topological complexity and automata-theoretic complexity measured by the Rabin-Mostowski index. My aim will be to prove certain results of strict containment using purely topological methods. At the end I hope to present some recent results about extensions of MSO.