Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zerodimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called ∆ 0 α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ≤ β < ω1 the degree-structure induced on X by the ∆ β -reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ≤ ω for every quasi-Polish space X, that αX ≤ 3 for quasi-Polish spaces of dimension 6= ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its