Consider the quotient set $\frac{\mathbb R} {\mathbb Q} $ obtained by the equivalence relation $$ x \equiv y \mod \mathbb Q \quad \text{iff} \quad x-y \in \mathbb Q $$ I was wondering if there exists a way to get a visual intuition of it. I mean, $\frac{\mathbb R} {\mathbb Z} $ can be thought of as the interval $[0,1) $. To this extent, what does $\frac{\mathbb R} {\mathbb Q} $ look like? And $\frac{\mathbb [0,1] } {\mathbb Q} $ (if defined in the same way)?
I know the example I provided is a bit different from what I asked, and there may not be an "obvious" or simple answer. I don't have any problem with the "abstract" definition, I was looking for a more "tangible" way to understand it. Thank you for your help, sorry if my answer seems useless or silly.
A way to represent $\mathbb R / \mathbb Q$ would be using the axiom of choice to pick up one element per class and to obtain a subset $A$ of the reals.
This is possible. Unfortunately, $A$ has a « bad behavior »! In particular $A \cap [0,1]$ is not Lebesgue measurable. See here for a proof.
So, representing $A$, for example graphically on the real line is not a simple mission...