3 research outputs found
Effective Wadge Hierarchy in Computable Quasi-Polish Spaces
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
Well Quasiorders and Hierarchy Theory
We discuss some applications of WQOs to several fields were hierarchies and
reducibilities are the principal classification tools, notably to Descriptive
Set Theory, Computability theory and Automata Theory. While the classical
hierarchies of sets usually degenerate to structures very close to ordinals,
the extension of them to functions requires more complicated WQOs, and the same
applies to reducibilities. We survey some results obtained so far and discuss
open problems and possible research directions.Comment: 37 page
A Q-Wadge Hierarchy in Quasi-Polish Spaces
The Wadge hierarchy was originally defined and studied only in the Baire
space (and some other zero-dimensional spaces). We extend it here to arbitrary
topological spaces by providing a set-theoretic definition of all its levels.
We show that our extension behaves well in second countable spaces and
especially in quasi-Polish spaces. In particular, all levels are preserved by
continuous open surjections between second countable spaces which implies e.g.
several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces. In fact,
many results hold not only for the Wadge hierarchy of sets but also for its
extension to Borel functions from a space to a countable better quasiorder Q