Understanding quotient topology

Perhaps the simplest interesting example is the quotient of $[0,1]$ obtained from the equivalence relation $E$ whose equivalence classes are the singletons $\{x\}$ for $0<x<1$ and the doubleton $\{0,1\}$. This identifies the endpoints $0$ and $1$ to a single point, and the quotient space is homeomorphic to $S^1$, the circle. Taking $S^1$ to be specifically the unit circle in the plane, one homeomorphism is the map

$$h:[0,1]/E\to S^1:p\mapsto\begin{cases} \langle\cos2\pi x,\sin2\pi x\rangle,&\text{if }p=\{x\}\\\\ \langle 1,0\rangle,&\text{if }p=\{0,1\}\;. \end{cases}$$

The quotient topology is exactly the one that makes the resulting space ‘look like’ the original one with the identified points glued together.

(This is really just the beginnings of an answer, because I’m not sure exactly what you want to know.)

When you are first learning about quotient spaces, it's difficult to get a visual intuition about them.

However, they're extremely important for cooking up new examples of topological spaces. And when you get used to the idea, you'll see the limitless potential they provide for making new spaces.

Getting used to them will only come from trying to understand all the examples that are written in your books.

Here are some ways that I've thought about quotients $q:E\rightarrow B$

Supposedly we already have a good understanding of the topology of $E$, and the quotient map $q$ is "gluing" parts of $E$ together to make a new space $B$. Our job is to understand $B$ at a level that we understood $E$.

Understanding $B$ from the inside: Example. If you lived in the interval $E=[0,1]$, then the numbers 0 and 1 would be like the start and finish of race track. Start at 0, go to 1, the end. However, if you now said 0=1 (in terms of points not numbers), then the finish of the race would also be the start of the race. 0 is your start go to 1=0 is your start go to 1... This is exactly what life would be like if you lived in the circle.

Understanding $B$ from the outside: Example, E is a rectangular stick of un-chewed bubble gum. To get B, Glue the top and bottom edges together to make a cylinder, then glue the other edges together i.e. the ends of the cylinder to make a torus. You have intuition already about a torus because you can hold it in your hand. So you can understand this quotient space from and extrinsic (outside) point of view. However, it's really best to understand the torus from the intrinsic (inside) point of view and to do that, start with the stick of gum E, and imagine you lived in the gum. Now, every time you traveled to the top of the gum, you're instantly transported to the bottom. Every time you made it to the left side, you are instantly transported to the right, etc.

Without this concept of gluing, it'd be pretty hard to quantitatively describe the torus and so the quotient concept is really handy. These examples are only the beginnings of what you can do

Once you get a good feeling for gluing points or subsets together to make a new space $B$, you'll need to understand the topology of this new space $B$ related to the topology old space $E$. This is where it's a good idea to let go of the need for visual intuition and to embrace definitions.