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?


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    $\begingroup$ Hint: the topology on this space is a "standard" topology that one learns as an example, upon first learning what a topological space is. And you can determine which topology it is by knowing that between every two irrational numbers there is a rational number, together with the definition of the quotient topology. $\endgroup$ – Ryan Budney Feb 27 '11 at 23:12
  • $\begingroup$ I think it is easier simply to use directly the fact that open sets in $\mathbb R/\mathbb Q$ are in bijections with the saturated open subsets of $\mathbb R$, that is, the open subsets $U\subseteq\mathbb R$ such that $$x\in U,q\in\mathbb Q\implies x+q\in U.$$ Can you describe all the saturated open subsets of $\mathbb R$? $\endgroup$ – Mariano Suárez-Álvarez Feb 28 '11 at 0:23
  • $\begingroup$ How do you want to take the quotient of $\mathbb{R}^2$ with respect to $S$? Since $S$ is not a subset of $\mathbb{R}^2$, I don't see a canonical way here. $\endgroup$ – Nemo Mar 27 '19 at 13:51

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?

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    $\begingroup$ Do you really mean $\mathbb{R}/S=\mathbb{R}/(\mathbb{R}/\mathbb{Q})$ and not $S=\mathbb{R}/\mathbb{Q}$? If so, how do you define $\mathbb{R}/(\mathbb{R}/\mathbb{Q})$? $\endgroup$ – joriki Jun 17 '11 at 5:50

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