The quotient topology is tricky! ...
There's a number of aspects to consider; all of them basically boil down to:
- Equip the quotient topology: $\pi:(X,\mathcal{S})\to Y$
- Check for quotient map: $\pi:(X,\mathcal{S})\to (Y,\mathcal{T})$
1.) Why do we take the quotient topology?
Given a plain space one wants to give it a topology that is good enough in the sense:
$g\text{ continuous}\iff g\circ\pi\text{ continuous}$
In other words, we don't want to artificially create new continuous maps plus we don't want to accidentally loose continuous maps.
Luckily, this can be guaranteed by imposing the quotient topology!
Given a plain space one wants to give it a topology that is good enough in the sense:
$f\text{ continuous}\iff\pi\circ f\text{ continuous}$
So one would like to maintain continuous maps with our plain space as codomain rather than as domain.
Unfortunately, this cannot be guaranteed in general by specific choice of some topology!
So that is all we can get!
(By the way, this is the same reason for the subspace topology as well as the product topology)
2.) How can we check and compare spaces for quotient topology?
a) Checking for quotient map:
Now, we are given already some topology. Is it the quotient topology?
Obviously, the projection must be necessarily continuous and surjective in order to be a quotient map, that is the given topology is the quotient topology.
However, this is not enough!!! (i.e. $\pi:[0,1)\to\mathbb{S}^1: s\mapsto \mathrm{e}^{2\pi\mathrm{i}s}$)
There's actually a number of lemmata providing sufficient conditions for it to be a quotient map, among them the one on saturated sets or the one on closed sets.
b) Comparing quotient maps:
Assume you are given now two topological spaces together with their own quotient maps and they seem to describe basically the same object, that is they seem to be isomorphic:
$\pi_1:(X,\mathcal{S})\to(Y_1,\mathcal{T}_1)$ and $\pi_2:(X,\mathcal{S})\to(Y_2,\mathcal{T}_2)$
Then, in order to prove that they are basically the same it suffices to check wether they make the same identifications:
$\pi_1(x)=\pi_1(\tilde{x})\iff\pi_2(x)=\pi_2(\tilde{x})$
(That is quite neat since one does not have to check topological properties)
2.)* How can we practically work with these?
There's a central result called the 'closed map lemma' that offers an astonishingly simple way to prove strong statements:
- Quotient map?
- Topological embedding?
- Homeomorphism?
All one has to check is continuity!!! (plus some little things)
In particular, this is quite useful in your case to prove the realization of the unit sphere as a quotient space. (Both of the preceding answers by Henno and by Aaron made use of this!)
Here's a draft how this goes:
$\pi:[0,1]\to[0,1]/\sim:s\mapsto [s]$ and $\epsilon:[0,1]\to\mathbb{S}^1:s\mapsto\mathrm{e}^{2\pi\mathrm{i}s}$
$\epsilon\text{ continuous and surjective}\Rightarrow\epsilon\text{ quotient map}$ ...with $[0,1]\text{ compact}$ and $\mathbb{S}^1\text{ Hausdorff}$
$\pi(s)=\pi(\tilde{s})\text{ iff }\epsilon(s)=\epsilon(\tilde{s})\Rightarrow[0,1]/\sim\cong\mathbb{S}^1$
... There's a lot more to discuss on quotient spaces but that should give you some first insights.