Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 329–359 | Cite as

Continuous reducibility and dimension of metric spaces

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Abstract

If (Xd) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (Xd) of positive dimension, there are uncountably many Borel subsets of (Xd) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.

Keywords

Continuous reducibility Wadge reducibility Borel sets Metric spaces 

Mathematics Subject Classification

03E15 54H05 

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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