TITLE: Quotient Space $\mathbb{R} / \mathbb{Q}$
QUESTION [14 upvotes]: I've just learned about topological quotient spaces and was wondering if anyone can help me with this example I thought of.
Let $(\mathbb{Q}, +)$ be the usual group of rational numbers for addition, likewise $(\mathbb{R}, +)$. Set $S$ to be the set of all cosets, t.i. $S=\mathbb{R}/\mathbb{Q}=\{x + \mathbb{Q} \mid x \in \mathbb{R} \}$. What is the quotient space $\mathbb{R} / S$ like? ($\mathbb{R}$ is equipped with the regular euclidian topology) What is it homeomorphic to? What does a typical open set look like?
Thanks.

REPLY [4 votes]: 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?