Quotient Space $\mathbb{R} / \mathbb{Q}$
Solution 1:
Since stackexchange is being silly and I can't seem to comment on my own question - I'll post this as an answer.
I'm thinking the topology is trivial on the set $S$. Since if the set $U$ is open in $\mathbb{R} / S$ then it's preimage of $q$ (where $q$ is quotient mapping) must be open in $\mathbb{R}$, meaning there exists an open interval $J \subseteq q^{-1}(U)$. But $q(J)$ equals all of the cosets in $\mathbb{R} / S$. Am I right?