In Topology 2ed by Munkres, the following definition is given of a quotient map:

Let $\displaystyle X,Y$ be topological spaces; let $\displaystyle p:{X}\rightarrow{Y}$ be a surjective map. The map $\displaystyle p$ is said to be aquotient mapprovided a subset $\displaystyle {U}\subset{Y}$ is open in $\displaystyle Y$ if and only if $\displaystyle {p^{-1}}(U)$ is open in $\displaystyle X$.

Then he gives the following example:

Let $\displaystyle X$ be the subspace $\displaystyle [0,1]\cup[2,3]$ of $\displaystyle \mathbb{R}$. The map $\displaystyle p:{X}\rightarrow{Y}$ defined by:

$\displaystyle p(x)=x$ if $\displaystyle x\in [0,1]$

$\displaystyle p(x)={(x-1)}$ if $\displaystyle x\in [2,3]$

In order to show that this map is a quotient map, one needs to show that is is surjective, continuous and closed. The first two are clear, however I am not so sure about how to verify that it is a closed map.

Also, he notes that the map is not open, saying as proof that the image of the "open set" $\displaystyle [0,1]$ of $\displaystyle X$ is not open in $\displaystyle Y$.

Can someone please tell me:

(i) how to verify that the map is a closed map

(ii) how $\displaystyle [0,1]$ can be considered open in $\displaystyle X$

(iii) how then is image of $\displaystyle [0,1]$ not open in $\displaystyle Y$