Is there an equivalence relation $R$ on $[0,1]\cap \mathbb{Q}$ such that $([0,1]\cap \mathbb{Q})/R$ is connected, Hausdorff, and has more than $1$ point?

$\begingroup$ Does $[0,1]\cap\mathbb{Q}$ inherit the subspace topology from $[0,1]$ in the interval topology? If so, I believe this is the same as the rational interval $[0,1]$ in the interval topology. $\endgroup$ – Alec Rhea Nov 12 '17 at 8:19

1$\begingroup$ @AlecRhea That's correct, $[0,1]\cap\mathbb{Q}$ inherits the subspace topology from $[0,1]$ with the interval topology (or Euclidean topology, which is the same here). $\endgroup$ – Dominic van der Zypen Nov 12 '17 at 8:49
Yes, there exists such a relation on $\mathbb Q$.
Just use the fact that the rational projective space $\mathbb QP^\infty$ from (the answer to) this question is a countable, Hausdorff, connected (and even topologically homogeneous). By definition, the space $\mathbb QP^\infty$ is a quotient (and even open) image of a countable metrizable space without isolated points. By the classical Sierpinski theorem such space is homeomorphic to $\mathbb Q$ (and to $\mathbb Q\cap[0,1]$, too). So, $\mathbb QP^\infty$ is a connected Hausdorff quotient (even open) image of $\mathbb Q$.

$\begingroup$ " from (the answer to) this question"  not only in the answer, in the question itself too:) $\endgroup$ – Fedor Petrov Nov 12 '17 at 20:39

$\begingroup$ @FedorPetrov Sorry. I had in mind that the notation $\mathbb QP^\infty$ appeared only in the answer but not in the question. That is why I wrote this way. $\endgroup$ – Taras Banakh Nov 12 '17 at 22:17