# 2013 seminar talk: Some Hierarchies of $\mathsf{QCB_0}$-Spaces

Talk held by Victor Selivanov (A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia and Siberian Branch of the Russian Academy of Sciences) at the KGRC seminar on 2013-10-17.

### Abstract

In this joint work with Mathias Schroeder we define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain $P_\omega$, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations (in the sense of computable analysis). We concentrate on the non-collapse property of the hierarchies, on the relationships between hierarchies in the two classes, and on the relationship with the Kleene-Kreisel continuous functionals.