7
$\begingroup$

Assume that $X$ is a metric space, and $\sim$ is an equivalence relation on $X$. Furthermore we assume that the number of elements in each equivalence class is bounded by a positive constant.

Does the quotient topology on $X/{\sim}$ and the topology induced by the Hausdorff-metric on $X/{\sim}$ coincide?

$\endgroup$
  • $\begingroup$ Do you want $X$ to be a metric space? $\endgroup$ – Ben McKay Jun 20 '16 at 13:57
  • $\begingroup$ Yes, of course. $\endgroup$ – Tim L. Jun 20 '16 at 14:00
11
$\begingroup$

Let $X=[0, \infty)$ with the metric $d(x,y)=|x-y|$. The equivalence relation is $x\sim y$ iff $x=y$ or $xy=1$.

In the Hausdorff metric on $X/{\sim}$, the open ball with radius $1/2$ around the equivalence class $\{0\}$ contains only $\{0\}$. However $\{0\}$ is not open in $X$, so $\{\{0\}\}$ is not open in the quotient topology on $X/{\sim}$.

$\endgroup$

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.