Borel subsets of the real line and continuous reducibility
Volume 244 / 2019
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 \emph {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 this 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 another main result, 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.
