In the first part of the talk I present a joint work with Matthias Schroeder where we extend the Luzin hierarchy of \(\mathsf\)-spaces introduced in in our previous work to all countable ordinals, obtaining in this way the hyperprojective hierarchy of \(\mathsf\)-spaces. We extend to this larger hierarchy all main results of the previous work. In particular, we extend the Kleene-Kreisel continuous functionals of finite types to the continuous functionals of countable types and relate them to the new hierarchy. We show that the category of hyperprojective \(\mathsf\)-spaces has much better closure properties than the category of projective \(\mathsf\)-space. As a result, there are natural examples of spaces that are hyperprojective but not projective.

In the second part I present a work in progress (joint with Matthew de Brecht and Matthias Schroeder) where we define and study new hierarchies based on the idea to classify \(\mathsf\)-spaces according to the complexity of their bases. The new hierarchies complement the previous ones and provide new tools to investigate non-countably based \(\mathsf\)-spaces. We concentrate on the non-collapse properties of the new hierarchies and on their relationships with the older ones. As a bi-product, we show that there is no universal \(\mathsf\)-space.