Understanding quotient topology

Question

Going through some wiki notes and books I found that a quotient space (also called an identification space) is, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. However, I am having trouble to understand these points. Also, I want to understand quotient topology on a given set. What is the motivation behind constructing quotient topology in a given set? I need simple explanation that can make me understand about quotient topology.

Thank you very much.

Answer 1

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.)

Answer 2

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.

Answer 3

I think as it is a very fundamental concept, so it is natural that many meaningful arguments for it exist from different angles. I would approach quotient topologies considering that they share some similarities with the quotient groups at least intuitively. So for me, the difference is mainly the difference between a group (set with operations) and a topology (set with structures).

Besides that, let $X$ and $R$ be topological spaces.

Not necessary, but it would help us to imagine if we assume they are Riemann manifoldness of some positive dimensions, distinguished (labeled) by a mark or boundary. Assuming that manifoldness is deformable through continuous or discrete (quanta) paths, Imagine two spaces are parametrized by two sets and each value of parameters results in a manifoldness that is accessible using an (equivalence) axioms (like what is the relation, criterion or standard of our labeling that is with certain magnitude of clarity $\mu$).

If everything we just said made some sense together, then we may define $Q=X/R$ and call it the quotient topological space that is the result of equivalence relations of $R$ but this time on $X$. More Loosely speaking, if the $R$ type manifoldness was lying somewhere on $X$ manifoldness previously and had some variability, it now shrank to a point of this new space and have no variability anymore.

Gluing in a quotient space may refer to a similar interpretation, that any manifoldness in $X$ subjected to $R$-equivalence, shrinks/glued to a single point. hence, it is a refinement of topology in being more specific and It is also a smaller space than $X$.

As for constructing a new (more specific) space, there is a connection to the concept of information for obvious reasons. Also, another point of view could be that we assume topological space as a generative system of a certain kind, many objects won't be generated in quotient space that would have been in an un-quotient space. (I am not sure if that's a thing but like $Q*R$)

Personally, I believe the quotient spaces take more energy to be visualized as daily geometric objects might not show these symmetry breaks imposed by certain logic that occur on a different scale. e.g. in quantum field theory etc. especially as $R$ getting closer to $X$ itself. For instance, imagine a Möbius strip as a quotient space of torus subjected to order equivalence relationship for inside and outside.

the manifoldness and variability concepts are here are from Riemann's Thesis

Answer 4

Let ${ X }$ be a topological space. Say we are collapsing some points of ${ X },$ i.e. are introducing an equivalence relation ${ \sim }$ on ${ X }.$

We can ask ourselves : Can we endow set ${ X /{\sim} }$ with a topology such that the collapse ${ X \overset{\pi}{\twoheadrightarrow} X /{\sim} }$ is a continuous transformation ?

Let ${ \tau }$ be a topology on ${ X /{\sim} }$ such that ${ X \overset{\pi}{\twoheadrightarrow} X/{\sim} }$ is continuous. Now ${ \tau }$ must be contained in ${ \lbrace U \subseteq X/{\sim} : \pi ^{-1} (U) \text{ is open in } X \rbrace }.$

But ${ \tau ^{\ast} }$ ${ := \lbrace U \subseteq X/{\sim} : \pi ^{-1} (U) \text{ is open in } X \rbrace }$ is itself a topology on ${ X /{\sim} },$ and under this ${ X \overset{\pi}{\twoheadrightarrow} X/{\sim} }$ is continuous.

So ${ \tau ^{\ast} }$ above is the largest topology on ${ X /{\sim} }$ under which ${ X \overset{\pi}{\twoheadrightarrow} X /{\sim} }$ is continuous. We call this the quotient topology on ${ X /{\sim} }.$