Towards a descriptive theory of cb0-spaces

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably basedT0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case ofk-partitions. In particular, we investigate the difference hierarchy ofk-partitions and the fine hierarchy closely related to the Wadge hierarchy.

WADGE HIERARCHY OF DIFFERENCES OF CO-ANALYTIC SETS

We begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets.

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 nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-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 Δ0β-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 ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.