Example of quotient map from Munkres book

Question

I was reading the notion of quotient map, topology and space but ran into the following example.

In this example I have understood almost everything except one moment: How to prove rigorously that $p(x)$ is closed map.

Would be thankful if anyone will show the rigorous proof.

Answer 1

If you want to do it from scratch, note that $f:x\mapsto x$ and $g:x\mapsto x-1$ are closed maps on $\mathbb R$ (this is easy to prove).

Let $C$ be closed in $X$, so there is a closed set $C'\subseteq \mathbb R$ such that

$X\cap C'=[0,1]\cap C'\sqcup [2,3]\cap C'=C.$ Then,

$p(C)=f([0,1]\cap C'))\sqcup g([2,3]\cap C'))=$

$[0,1]\cap C'\sqcup [1,2]\cap g(C')=[0,2]\cap (C'\cup g(C'))$.

Since $C'$ is closed in $\mathbb R$ by assumption and $g(C')$ is closed in $\mathbb R$ also, by the first remark, we conclude that $p(C)=[0,2]\cap (C'\cup g(C'))$ is closed in $Y$.

Answer 2

Any closed subset of $[0,1] \cup [2,3]$ is compact. Since $p$ is continuous its image is compact, hence closed.

Answer 3

The map $p$ restricted to each interval is just a homeomorphism so closed. From this we conclude that this combined map is also closed.