13
$\begingroup$

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.

  • 5
    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.2011-02-27
  • 0
    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 $\mathbb R$?2011-02-28

1 Answers 1

4

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?

  • 1
    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 $\mathbb{R}/(\mathbb{R}/\mathbb{Q})$?2011-06-17