Properties of Cantor set

Question

$[0,1]$ is not homeomorphic to $[0,1]×[0,1]$ but $C$ is homeomorphic to $C \times C$ where $C$ is the Cantor set.

I know both the proof. I am asking which property of $C$ is the reason of this absurdity!

Answer 1

$C$ is the (up to homeomorphism) unique zero-dimensional (or totally disconnected) compact metric space without isolated points.

If $X$ is such a space, so is $X^n$ for any $n$: still compact, totally disconnected, metric and no isolated points so it's homeomorphic to $X$.

Other spaces with unique charaterisations also have such preservations by finite products:

• $\mathbb{Q}$: the unique countable metric space without isolated points.
• $\mathbb{P}$ (the irrationals in the reals): the unique completely metrisable zero-dimensional separable metric space that is nowhere locally compact (i.e. the interior of any compact subset is empty).
• $C\setminus \{0\}$ (the Cantor set minus a point), the unique locally compact non-compact separable metric space that is totally disconnected.

And some more exist too. All of the above are homeomorphic to their squares.