Wadge-like reducibilities on arbitrary quasi-Polish spaces. (10th November 2014) - MOTTO ROS, LUCA; SCHLICHT, PHILIPP; SELIVANOV, VICTOR
Abstract : The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-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 ⩽ β < ω1 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 ⩽ ω for every quasi-Polish space X, that α X ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.