  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. We concentrate on the
non-collapse property of the hierarchies and on the relationships between
hierarchies in the two classes.
