# Intuition behind quotient topology

Extracted from my text:

We see that the interval $\left [ 0,1 \right ]$ becomes the circle $S^{1}$when we $\textit{glue}$ the points 0 and 1. 0 and 1 are both thought of as a $\textit{single point}$. More formally, this defines an $\textit{equivalence relation }$ ~ on $\left [ 0,1 \right ]$ in which x~x for every x, 0~1 and 1~0.

The circle is then the collection of $\textit{equivalence class} \left [ x \right ]=\left \{ y:y~x \right \}$, each of which is a single point $x \in \left ( 0,1 \right )$ or the pair $\left \{ 0,1 \right \}$

However, I do not quite understand the part : "each of which is a single point $x \in \left ( 0,1 \right )$ or the pair $\left \{ 0,1 \right \}$

Would someone kindly explain?

It is saying that every equivalence class is made up of one exact point, up to the tuple $0,1$.

This is because of how the equivalence relation is defined: $x\sim x,1\sim 0,0\sim 1$. To be more exhaustive:

• if $x\neq 0,1$ then $[x]=\{y\mid y\sim x\}=\{x\}$

• if $x=0$ then $[0]=\{y\mid y\sim 0\}=\{0,1\}$, because $0\sim 0$ and $0\sim 1$

• similarly for $x=1$

• More precisely, $x \sim y$ if and only if $x=y$,$(x,y)=(0,1)$ or $(x,y)=(1,0)$.
– Ian
Commented Sep 13, 2016 at 13:05
• Yes. Or equivalently $R=\Delta\cup(0,1)\cup(1,0)$. Commented Sep 13, 2016 at 13:08