Wadge-like reducibilities on arbitrary quasi-Polish spaces

TL;DR

This paper investigates the complexity of reducibility hierarchies on quasi-Polish spaces, establishing bounds on when these hierarchies become simple, extending Wadge theory beyond zero-dimensional spaces.

Contribution

It introduces bounds on the complexity of reducibility hierarchies on quasi-Polish spaces, showing that these hierarchies simplify at low levels of the b4^0_eta-reductions, with bounds proven to be optimal.

Findings

b4^0_eta-reduction hierarchies are simple for b4^0_eta with eta \u2264 3 in many spaces

For all quasi-Polish spaces, the hierarchy simplifies at eta \u2264 

The bounds are optimal for spaces like the real line and its powers.

Abstract

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.