Metrizability of quotient spaces of metric spaces

Question

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.

Answer 1

If $X$ is a standard flat torus and the equivalence relation declares two points to be equivalent if they lie on the same line with a (fixed) irrational slope, the quotient is clearly not metrizable.

Answer 2

A very simple way in which the quotient can fail to be metrizable is if there are equivalence classes that are not closed. Take your original space to be $\mathbb{R}$ and let $\sim$ have as its equivalence classes $(0,1)$ and all singletons $\{x\}$ with $x\notin(0,1)$. In the quotient topology, you can not separate $(0,1)$ and $1$ by open sets and the quotient space fails to be Hausdorff.