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?



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


