= Solution
\b[False.] Take the <metric space> $X=\mathbb R\times\{0,1\}$ with the <Euclidean distance> and <subspace topology> inherited from $\mathbb R^2$. For every $t\neq0$, identify $(t,0)$ with $(t,1)$; leave the two points $(0,0)$ and $(0,1)$ in separate <equivalence classes>. All <equivalence classes> have one or two points, so all are <closed> in this <metric space>.
Let $o_0,o_1$ be the two distinct images of the origins. If $U_0,U_1$ are any <open> <neighbourhoods> of $o_0,o_1$ in the <quotient space>, their <preimages> contain respectively
$$
(-\varepsilon,\varepsilon)\times\{0\},\qquad(-\delta,\delta)\times\{1\}
$$
for some $\varepsilon,\delta>0$. Choose $0<t<\min(\varepsilon,\delta)$. The common image $p(t,0)=p(t,1)$ belongs to both $U_0$ and $U_1$. Therefore the origins have no disjoint <open> <neighbourhoods> and the <quotient space> is not <Hausdorff>. This is the <line with two origins>.
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