Munkres defines a quotient maps using open sets (as is done here.) Then he comments that one could also formulate the definition using closed sets instead.
He considers the following example to show a function that is a quotient map, but not on the quotient map's restriction on a subset of the domain:
Let $X$ be the subspace $[0.1] \cap [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0.2]$ of $\mathbb{R}$. The map $p:X \rightarrow Y$ defined by $$ p(x) = \begin{cases} x & x \in [0,1] \\ x-1 & x \in [2,3] \end{cases} $$ is readily seen to be surjective, continuous, and closed. Hence it is a quotient map. Now consider $[0,1) \cup [2,3] = A \subset X$, then the map $p|_A: A \rightarrow Y$ is continuous and surjective but not a quotient map. To see this, $[2,3]$ is open in $A$ and is saturated with respect to $p|_A$, but its image is not open in $Y$.
I see that this is true, but isn't $[2,3] \subset A$ also closed so that the image is closed in $X$. As far as I can see, $p|_A$ can be a closed map - isn't this true?
$p\mid_A$ is not a closed map. $[0,1)$ is closed in $A$ but $p|_A([0,1)) = [0,1)$ is not closed in $Y$.