Title: Borel subsets of the real line and continuous reducibility,
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.