A Suslin line is defined as a non-empty, complete, dense, linear order without endpoints and in which every collection of disjoint intervals is countable. A Suslin tree on the other hand, is an uncountable tree with no uncountable chains or antichains.
An exercise asks us to show that you could construct one from the other; I managed to get the tree from the line, but I can't even seem to get started the other way. A hint says to construct the line from maximal chains of the tree.
Intuitively, I think I understand why this is: the maximal chains are totally ordered, so you have a "nice" way of comparing things within a chain. So say that I declare that my Suslin line $S$ is the union of all the maximal chains of the Suslin tree $T$. I now need to linearly order $S$ somehow, but I'm not seeing how to compare two elements that are in different chains.
Am I thinking about this the right way? Any help is appreciated!
HINT: You aren’t going to compare things within the chains: the elements of the Suslin line will be the maximal chains themselves, not their elements. The basic idea is to let $\prec$ be an arbitrary linear order on $T$. Let $C_0$ and $C_1$ be distinct maximal chains in $T$; there must be a minimal $\alpha<\omega_1$ such that $C_0(\alpha)\ne C_1(\alpha)$. Now define $C_0<C_1$ iff $C_0(\alpha)\prec C_1(\alpha)$. (Thus, $<$ is a bit like a lexicographic order.) Let $\mathscr{C}$ be the set of maximal chains; then $\langle\mathscr{C},<\rangle$ is a Suslin line (though you may have to tinker a little with $T$ to ensure this).