= Solution
\b[False.] Give $X=\mathbb R$ its usual <topology>, identify all points of $(0,1)$ with one another, and leave every other point in a <singleton> <equivalence class>. Take the <closed set> $K=\{1/2\}$.
Its image $p(K)$ is a <singleton> in the <quotient space>, but its full <preimage>, also called the <saturation of a subset>, is
$$
p^{-1}(p(K))=(0,1).
$$
This is not <closed> in $\mathbb R$, since $0$ belongs to its <closure>. In the <quotient topology>, a <subset> is <closed> exactly when its <preimage> is <closed>, by applying the defining open-set rule to complements. Thus $p(K)$ is not <closed>, and this <quotient map> is not a <closed map>.
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