Continuous reducibility for the real line

We study Borel subsets of the real line up to continuous reducibility. We firstly show that every quasi-order of size ω1 embeds into the quasi-order of Borel subsets of the real line up to continuous reducibility. We then prove that at least all the types of gaps in P(ω)/fin appear and determine several cardinal characteristics of this quasi-order. We also begin an analysis of the Fσ subsets of the real line by characterizing the sets reducible to Q and constructing the least non-trivial set below Q