= Solution
{c}
For the quotient map $q:X\to X/{\sim}$, the <quotient topology> is defined by
$$
U\subseteq X/{\sim}\text{ open}\iff q^{-1}(U)\text{ open in }X.
$$
Take the <Hausdorff space> $X=\mathbb R$ and identify two numbers precisely when they are both rational or both irrational. The quotient has two points. The inverse images of its singleton subsets are $\mathbb Q$ and $\mathbb R\setminus\mathbb Q$, neither of which is open in $\mathbb R$, since each is dense and has dense complement. Hence the quotient has the <indiscrete topology> and is not <Hausdorff>.
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