Continuous reducibility and dimension of metric spaces

Archive for Mathematical Logic 57 (3-4):329-359 (2018)
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If 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 of positive dimension, there are uncountably many Borel subsets of 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 \\) is called the Wadge quasi-order for \\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \\), is a well-quasiorder if and only if \\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.



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Baire reductions and good Borel reducibilities.Luca Motto Ros - 2010 - Journal of Symbolic Logic 75 (1):323-345.
Borel-amenable reducibilities for sets of reals.Luca Motto Ros - 2009 - Journal of Symbolic Logic 74 (1):27-49.
More on Wadge determinacy.Alessandro Andretta - 2006 - Annals of Pure and Applied Logic 144 (1-3):2-32.
A Wadge hierarchy for second countable spaces.Yann Pequignot - 2015 - Archive for Mathematical Logic 54 (5-6):659-683.
Beyond Borel-amenability: scales and superamenable reducibilities.Luca Motto Ros - 2010 - Annals of Pure and Applied Logic 161 (7):829-836.

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