Example 1, Sec. 22, in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed? 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? 
 A: You can avoid compactness by taking advantage of the very simple nature of $p$. Let $p_0=p\upharpoonright[0,1]$ and $p_1=p\upharpoonright[2,3]$; then $p_0$ and $p_1$ are homeomorphisms of $[0,1]$ onto itself and of $[2,3]$ onto $[1,2]$, respectively. In particular, $p_0$ and $p_1$ are closed maps from $[0,1]$ to itself and from $[2,3]$ to $[1,2]$
Let $F$ be a closed subset of $X$. Let $F_0=F\cap[0,1]$ and $F_1=F\cap[2,3]$; then 
$$p[F]=p[F_0]\cup p[F_1]=p_0[F_0]\cup p_1[F_1]\;.$$
$F_0$ is closed in $[0,1]$, so $p[F_0]$ is closed in $[0,1]$ and therefore in $\Bbb R$. Similarly, $F_1$ is closed in $[2,3]$, so $p_1[F_1]$ is closed in $[1,2]$ and hence in $\Bbb R$. Thus, $p[F]=p_0[F_0]\cup p_1[F_1]$ is closed in $\Bbb R$ and hence in $Y$.
For the other question, recall that $S\subseteq A$ is saturated with respect to $q$ iff $q^{-1}\big[q[S]\big]=S$. Here $q\big[[2,3]\big]=[1,2]$, and $q^{-1}\big[[1,2]\big]=[2,3]$, so $[2,3]$ is $q$-saturated.
A: Since $X$ is compact any continuous map from $X$ to a Hausdorff space is closed.
About the second question $q^{-1}([1,2])=[2,3]$, so $[2,3]$ is saturated with respect to $q$
