TITLE: Consider $\mathsf{A,B,C},$ $\dots$ in a sans serif font. Each of these gives a graph in the plane. Sort these into homeomorphism classes.
QUESTION [9 upvotes]: Problem:
Show that homeomorphism is an equivalence relation on topological spaces. Now consider the capital letters of the alphabet $\mathsf{A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z},$ in a sans serif font. Each of these gives a graph in the plane. Sort these into homeomorphism classes. (The partition may depend on the font! In particular, $\mathsf{K}$ can be tricky.)

What I got so far:
$\mathsf{C\cong G\cong I \cong J \cong L \cong U \cong V \cong Z}$ 
They are not closed curves and can be curved and bent to one another.
$\mathsf{D\cong O}$
Closed curves that can easily be bent to one another.
$\mathsf{C \ncong O}$ since removing a point in $\mathsf{C}$ makes it disconnected while $\mathsf{O}$ stays connected.
$\mathsf{E\ncong O}$ for the same reason. 
$\mathsf{E\cong F}$
The standing line on $\mathsf{E}$ can be shrunk and the bottom line rotating and shrinking to make the straight standing line for $\mathsf{F}$. And the same reverse process for $\mathsf{F}$, which are homeomorphisms.
$\mathsf{K\cong X}$, this is easy to see.
and $\mathsf{K \ncong H}$ since deleting a point in $K$ gives 4 components but not in $H$.
I don't see any congruent letters for $\mathsf{A}$ and $\mathsf{H,R}$, so I think each is alone. 
$\mathsf{P\cong Q}$ is clear. 
$\mathsf{E\cong F\cong T \cong Y}$ as well.
$\mathsf{M\cong N\cong W}$ is also easy to see.
Is this correct, I would appreciate any corrections.

REPLY [3 votes]: Homeomorphisms allow twisting and bending, but not (permanent) cuts. You can think of connectedness here to decide when two letters are not homeomorphic (e.g., remove one point from each letter, etc.). I won't do everything here as I think that this exercise is very important to get a feeling for these things, but here's some examples:
$${\sf C}\cong {\sf I} \cong {\sf V}, \quad {\sf D} \cong {\sf O}, \quad {\sf A}\not\cong {\sf O}, \quad {\sf L}\not\cong {\sf D}.$$
This classification can be also done thinking of homotopy classes instead of homeomorphism classes. I don't think it helps trying to write explicit homeomorphisms here. I mean, what are you going to do, parametrize the letter? The best we can gain here is intuition. 
I refer you to the first chapter of A First Course in Algebraic Topology - Czes Kosniowski. It is a very nice book and may help you get this.