# Does the topology induced by the Hausdorff-metric and the quotient topology coincide?

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?

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

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 $X/{\sim}$.