Interesting theorems/facts about identification spaces I am now studying algebraic topology (still at the beginning). I am now studying identification spaces, adjunction spaces,... As I still don't know how these concepts are going to be used, I think I am getting a bit less interested. I would like you to present to me interesting theorems  about identification maps or theorems (not necessarily about identification maps) which are easier to understand, prove using the concept of identification maps. I think good examples of these would keep me motivated.
I am not looking for basic facts such as the composition of two identification maps is an identification map or other easy facts.
I realize that this question might be ambiguous, thus I don't mind deleting it if I don't get a response 
 A: A lot of the "fun" spaces are just idenfitication spaces on very simple spaces like squares and cubes. For example, the Klein bottle, the Möbius strip, the sphere, and the torus are all viewable as identification spaces on a simple square with interior. 
The circle can be seen as the identification space of the real line with $x\sim y$ if and only if $x-y$ is an integer.
Identification spaces are how we build stuff up out of smaller spaces - whenever we "glue" two spaces together, what we are really doing is creating an identification space on the disjoint union of the spaces.
A: A common source of (particularly nice) identification spaces comes from group actions on topological spaces.
Given any set $X$ and a group $G$, a group action of $G$ on $X$ is a function from $G\times X\rightarrow X$, denoted by $(g,x) \mapsto g\cdot x$ which is required to satisfy two axioms:


*

*$e\cdot x = x$ where $e\in G$ is the identity element and $x\in X$ is anything.

*$(gh)\cdot x = g\cdot(h\cdot x)$
In the case where $X$ is a topological space, we further require that for every $g\in G$, the function $\phi_g:X\rightarrow X$ given by $\phi_g(x) = g\cdot x$ be a homeomorphism.
Now, we can define an equivalence relation on $X$ by the following:  $$x\sim y \iff g\cdot x = y \text{ for some } g\in G.$$
Bullet point 1 tells us that $\sim$ is reflexive.  Using $g^{-1}$ shows that $\sim$ is symmetric, and bullet point 2 tells us that $\sim$ is transitive.  All that together just means that $\sim$ really is an equivalence relation.
One can then consider the identification set $X/\sim$.  It turns out, if $X$ is a topological space, the condition that all of the $\phi_g$ functions are homeomorphism is exactly what we need in order to put a topology on $X/\sim$ in such a way that the natural projection $\pi:X\rightarrow X/\sim$ is continuous.
What are some examples of this?  For starters, every example (except for the sphere) in Thomas Andrews's first paragraph is of this form for an action of $G = \mathbb{Z}$ or $\mathbb{Z}^2$ on $X=\mathbb{R}^2$.  His second paragraph is also of this form for $G = \mathbb{Z}$ and $X=\mathbb{R}$.
But there are many more examples.  If you've seen covering spaces, the fundamental group of a space acts on its universal covering, giving back the space as the quotient.  An example of this:  $G=\mathbb{Z}/2\mathbb{Z}$ acts on the sphere $S^n$ and the quotient is $\mathbb{R}P^n$.  Relatedly, $G = S^1$ (thought of as the group of unit complex numbers) acts on $S^{2n+1}$ with quotient $\mathbb{C}P^n$.
A: $B^n/S^{n-1}\cong S^n$ is pretty interesting to me.
