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

Raphaël Carroy; Luca Motto Ros; Salvatore Scamperti

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arXiv · 2302.01319

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

Raphaël Carroy·Luca Motto Ros·Salvatore Scamperti

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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.

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BibTeX RIS

Raphaël Carroy, Luca Motto Ros, Salvatore Scamperti. 2023-02-02. A classification of the Wadge hierarchies on zero-dimensional Polish spaces. https://arxiv.org/abs/2302.01319

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