  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.
