Continuous reducibility and dimension of metric spaces

TL;DR

This paper investigates the structure of Borel subsets under continuous reducibility in metric spaces, revealing that positive dimension spaces contain uncountably many incomparable subsets, unlike zero-dimensional spaces.

Contribution

It establishes that the Wadge quasi-order is a well-quasiorder if and only if the space has dimension zero, and introduces a new technique based on graph colorings for analyzing reducibility.

Findings

Uncountably many incomparable Borel subsets in positive dimension spaces

Wadge quasi-order is a wqo iff the space has dimension zero

New applications of graph coloring techniques

Abstract

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,\tau)$ is called the \emph{Wadge quasi-order} for $(X,\tau)$. We further show that this quasi-order, restricted to the Borel subsets of a Polish space $(X,\tau)$, is a \emph{well-quasiorder (wqo)} if and only if $(X,\tau)$ has dimension $0$, as an application of the main result. Moreover, we give further examples of applications of the technique, which is based on a construction of graph colorings.