Uncountable set of irrational numbers such that no two elements sum to a rational I'm trying to find such a set as described in the title to prove that the following metric space is separable:
$([0,1],d)$
where
$ d(x,y) = \left\{
 \begin{array}{ll}
  |x-y|  & \mbox{if } x-y \in \mathbb{Q}\\
  5 & \mbox{if } x-y \not\in \mathbb{Q}
 \end{array}
\right.$
If there exists an easier approach then any suggestions are welcome. Also if I'm going the wrong way and it is separable I would like to ask how to do it the other way around.

Note that $x\sim y\iff x-y\in\Bbb Q$ defines an equivalence relation on $[0,1]$. Moreover, this equivalence relation has uncountably many equivalence classes. Suppose that $V\subseteq[0,1]$ is a set which meets every equivalence class exactly on one point, and now consider $A=\{B(x,1/3)\mid x\in V\}$.
Show that $A$ is an uncountable set of pairwise disjoint open sets.
