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Wadge-like reducibilities on arbitrary quasi-Polish spaces

Published online by Cambridge University Press:  10 November 2014

LUCA MOTTO ROS ,
PHILIPP SCHLICHT  and
VICTOR SELIVANOV
LUCA MOTTO ROS
Affiliation:
Abteilung für Mathematische Logik, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, D-79104 Freiburg im Breisgau, Germany Email: luca.motto.ros@math.uni-freiburg.de
PHILIPP SCHLICHT
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Email: schlicht@math.uni-bonn.de
VICTOR SELIVANOV
Affiliation:
A. P. Ershov Institute of Informatics Systems, Siberian Division Russian Academy of Sciences, Novosibirsk, Russia Email: vseliv@iis.nsk.su
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Abstract

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

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