Let $X$ be the subspace $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0,2]$ of $\mathbb{R}$. The map $p \colon X \to Y$ defined by $$ p(x) \colon= \begin{cases} x \ &\mbox{ for } \ x \in [0,1], \\ x-1 &\mbox{ for } \ x \in [2,3] \end{cases} $$ is surjective and continuous. This much is clear to me.
But Munkres asserts that this map is also a closed (and hence a quotient) map. How to verify this?
Now if $A$ is the subspace $[0,1) \cup [2,3]$ of $X$, then the map $q \colon A \to Y$ obtained by restricting $p$ is continuous and surjective. Alright!
Now what stumps me is the following assertion by Munkres:
The set $[2,3]$ is saturated with respect to $q$. How to show this?