A classification of the Wadge hierarchies on zero-dimensional Polish spaces

TL;DR

This paper classifies all Wadge hierarchies on zero-dimensional Polish spaces using countable ordinals, revealing the complexity of their isomorphism problem from a descriptive set-theoretic perspective.

Contribution

It provides a complete, explicit classification of Wadge hierarchies on zero-dimensional Polish spaces up to order-isomorphism, using simple invariants.

Findings

Classification uses countable ordinals as invariants.

No Borel procedure exists to determine hierarchy isomorphism.

Results depend on explicit descriptions based on topological properties.

Abstract

We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $\aleph_1$-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.