Wadge-like reducibilities on arbitrary quasi-Polish spaces
arXiv:1204.5338 · doi:10.1017/S0960129513000339
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 Δ^0_α-reductions, and try to find for various natural topological spaces X the least ordinal α_X such that for every α_X \leq β< ω_1 the degree-structure induced on X by the Δ^0_β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that α_X \leq ω for every quasi-Polish space X, that α_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.
50 pages, revised version, accepted for publication on Mathematical Structures in Computer Science