A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff-Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Automated summary

The Wadge hierarchy, originally defined and studied in the Baire space, has been extended to arbitrary topological spaces by providing a set-theoretical definition of all its levels. The extension behaves well in second countable spaces and especially in quasi-Polish spaces. The levels are preserved by continuous open surjections between second countable spaces, leading to several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces.