There's a rather simple proof for the following theorem:
There exists an equivalence relation $\sim$ on the unit interval $I=[0,1]$ such that the quotient $I/{\sim}$ is homeomorphic to the unit square $I^2$.
We first form a surjective map $f\colon I\to I^2$ which is known to exist by several constructions. We then define $x\sim y$ for $x,y\in I$ if and only if $f(x)=f(y)$. The corresponding map $\tilde{f}\colon I/{\sim}\to I^2$ is then, by construction, a continuous bijection from a compact space to a Hausdorff space and so is a homeomorphism.
My question is in trying to get some kind of intuition for this equivalence $\sim$ if at all possible. Given the usual complexity in the definition of space filling curves, it seems like this might be a daunting task. I wonder if anyone could offer some insight - perhaps even just how large the preimage sets under $f$ can be, as I'm not even sure if these sets are finite. If these preimage sets are infinite, then I assume they are either densely embedded in $I$ or have the structure of a Cantor set - does this seem reasonable?
