Non Hausdorff quotient spaces

Inspired from K. Jänich, Topologie, Springer.


Let $X$ be Hausdorff. What are the requirements for $\sim$ that the quotient space $X / \sim$ becomes Hausdorff?


If $X / \sim$ is Hausdorff, then all equivalence classes of $\sim$ in $X$ are closed: If there was a $y \in \overline{[x]} \setminus [x]$, then one could not seperate $[y]$ from $[x]$ in $X / \sim$.

Does the converse hold?

Are there equivalence relations $\sim$ on a Hausdorff space $X$ where the sets $[x]$ are closed, but $X / \sim$ is not Hausdorff? For that consider the two examples:
The following applet visualizes differerent topologies in $\mathbb{R}^2/\sim$. The orange shape corresponds to an open neighborhood of $[x]$ in the given topology.

Select one topology of the following list: