I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (for the cantor set $C$). And then I got stuck, when I considered the rationals. So the question is:

Is $\mathbb{Q}^2$ homeomorphic to $\mathbb{Q}$ ?

Yes, Sierpinski proved that every countable metric
space without isolated points is homeomorphic to the rationals:
http://at.yorku.ca/p/a/c/a/25.htm .

An amusing consequence of Sierpinski's theorem is that
$\mathbb{Q}$ is homeomorphic to $\mathbb{Q}$. Of course here one
$\mathbb{Q}$ has the order topology, and the other has the $p$-adic
topology (for your favourite prime $p$) :-)
