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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.