A Q-Wadge Hierarchy in Quasi-Polish Spaces

TL;DR

This paper extends the Wadge hierarchy to quasi-Polish spaces, providing a set-theoretic framework that preserves hierarchy levels under continuous open surjections, with implications for Borel functions and better quasiorders.

Contribution

It introduces a set-theoretic extension of the Wadge hierarchy to quasi-Polish spaces, maintaining its properties and applicability beyond zero-dimensional spaces.

Findings

Hierarchy levels are preserved in second countable spaces.

The extension behaves well under continuous open surjections.

Results apply to Borel functions into countable better quasiorders.

Abstract

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). We extend it here 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-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.