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?



Does $[0,1]\cap \mathbb{Q}$ have a connected $T_2$ quotient?
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](https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space/285556#285556) 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 $\mathbb Q$.


