Towards a Descriptive Theory of cb_0-Spaces

TL;DR

This paper extends classical Descriptive Set Theory results to a broad class of cb_0-spaces, exploring hierarchies of sets and partitions, including the Wadge hierarchy, to deepen understanding of their structure.

Contribution

It introduces a generalized framework for Descriptive Set Theory applicable to cb_0-spaces, analyzing hierarchies of sets and partitions beyond traditional spaces.

Findings

Extended Borel, Luzin, and Hausdorff hierarchies to cb_0-spaces

Analyzed the difference hierarchy of k-partitions

Explored the fine hierarchy related to the Wadge hierarchy

Abstract

The paper tries to extend results of the classical Descriptive Set Theory to as many countably based T_0-spaces (cb_0-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.