Bad Wadge-like reducibilities on the Baire space

Tom 224 / 2014

Luca Motto Ros Fundamenta Mathematicae 224 (2014), 67-95 MSC: Primary 03E15; Secondary 03E60, 03D55, 03D78, 54C10, 54E40. DOI: 10.4064/fm224-1-4


We consider various collections of functions from the Baire space $ {}^{\omega } \omega $ into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on $ {}^{\omega } \omega $ (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of $ {}^{\omega } \omega $, and we show that the resulting hierarchies of degrees are much more complicated than the classical Wadge hierarchy; in particular, they always contain large infinite antichains, and in most cases also infinite descending chains.


  • Luca Motto RosMathematisches Institut – Abteilung für Mathematische Logik
    Albert-Ludwigs-Universität Freiburg
    Eckerstraße 1
    D-79104 Freiburg im Breisgau, Germany

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