Effective Wadge Hierarchy in Computable Quasi-Polish Spaces

TL;DR

This paper develops an effective version of the Wadge hierarchy for computable quasi-Polish spaces, extending classical results and analyzing hierarchies of sets and partitions relevant for computable analysis.

Contribution

It introduces an effective Wadge hierarchy in computable quasi-Polish spaces and extends the effective Hausdorff-Kuratowski theorem to these spaces and k-partitions.

Findings

Hierarchy levels are preserved by computable effectively open surjections.

Effective Hausdorff-Kuratowski theorem holds in all computable quasi-Polish spaces if it holds in Baire space.

Extended the theorem to k-partitions.

Abstract

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of k-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to k-partitions.