Borel subsets of the real line and continuous reducibility

TL;DR

This paper investigates the structure of Borel subsets of the real line under continuous reducibility, revealing complex embeddings, non-structure phenomena, and characterizations of certain reducible sets.

Contribution

It introduces new structural results about Borel sets under continuous reducibility, including embeddings of complex posets, non-existence of complete sets, and characterizations of reducibility to rationals.

Findings

Embeddings of complex posets into Borel classes above open and closed sets.

No complete sets exist for these Borel classes.

Characterization of $F_\sigma$ sets reducible to $Q$ and the construction of a minimal such set.

Abstract

We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order $\omega_1$, its reverse $\omega_1^\star$ and the poset $\mathcal{P}(\omega)/\mathsf{fin}$ of inclusion modulo finite error. As a consequence of its proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to $\mathbb{Q}$. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In our second main theorem, we introduce a combinatorial property that is shown to characterize those $F_\sigma$ sets that are reducible to $\mathbb{Q}$. This is applied to construct a minimal set below $\mathbb{Q}$ and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.