Towards a descriptive theory of cb₀-spaces

Abstract

The paper tries to extend results of the classical Descriptive Set Theory to as many countably based T0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we consider also the more general case of k-partitions. In particular, we investigate the difference hierarchy of k-partitions and the fine hierarchy closely related to the Wadge hierarchy.

Well-Quasi Orders and Hierarchy Theory

“…, {k − 1} by the operations and + β for all 1 ≤ β < α. The extension of Theorem 4.4 could probably be obtained by defining suitable iterated versions of the h-quasiorder in the spirit of [111]. Since B(k N ) = α<ω1 ∆ 0 α (k N ), we obtain the characterization of Wadge degrees of Borel k-partitions.…”

“…By a result in [111], in the case k = 2 the QO T k (ω) is semi-well-ordered (in fact, T 2 (ω 1 ) is also semi-well-ordered). This indicates a possible relation to the Wadge hierarchy from Section 2.5.…”

“…The structure (T ⊔ k (ω); ≤ h ) may be expanded by natural operations inducing a rich algebraic structure on the quotient-poset. These operations, introduced and studied in [108,109,111], are important for relating the h-QOs to hierarchy theory.…”

“…Let us briefly recall from [111] some operations on the iterated labeled forests and collect some of their properties used in the sequel. The ω-ary operation of disjoint union on T ⊔ k (ω 1 ) is defined in the obvious way.…”

“…Although this result gives an important information about the Wadge degrees of Borel k-partitions, it is far from a characterization. Here we briefly discuss some steps to such a characterization made in [35,104,111,112]). The approach of [104,112] is to characterize the initial segments (∆ 0 α (k N ); ≤ W ) for bigger and bigger ordinals 2 ≤ α < ω 1 .…”