1

Suppose that $C$ is the cantor set and let $\sim$ be an equivalence relation on $C$. Are there necessary and sufficient conditions on $\sim$ to guarantee that the quotient $C/\sim$ is Hausdorff?

What I know:

  • If $q: X \to Y$ is an arbitrary quotient map then $Y$ is $T_1$ if and only if the fibres of $q$ are closed. So the equivalence classes of $\sim$ must be closed in $C$.

  • Since $C$ is compact and normal, the results listed in https://math.stackexchange.com/a/1697569/134117 imply that if the quotient map $q: C \to C/\sim$ is closed then $C/\sim$ will be Hausdorff. What conditions can be put on $\sim$ to guarantee that the quotient map is closed?

Zorngo
  • 1,421
  • I'm pretty sure $X/!!\sim$ is Hausdorff iff $\sim$ is closed as a subset of $X\times X$. You can try to prove that – Luiz Cordeiro May 25 '18 at 02:55
  • Ok, that's been asked before... https://math.stackexchange.com/questions/91639/x-sim-is-hausdorff-if-and-only-if-sim-is-closed-in-x-times-x No conditions on $X$ necessary, apparently – Luiz Cordeiro May 25 '18 at 02:57

0 Answers0