Splitting and non-splitting in the difference hierarchy

In this paper, we investigate splitting and non-splitting properties in the Ershov difference hierarchy, in which area major contributions have been made by Barry Cooper with his students and colleagues. In the first part of the paper, we give a brief survey of his research in this area and discuss a number of related open questions. In the second part of the paper, we consider a splitting of 0′ with some additional properties.

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 hardness in Scott spaces and its effectivization

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.