Show the Cantor set $C$ is equal to its product $C\times C$. I've been reading up on the Cantor set, and it is simple to show a bijection $C \to [0,1]$.
I was thinking that it would be easy to show that there exists some space filling curve by showing that $C = C \times C$ (equality in the sense of homeomorphism). Because then I would have $[0,1] = [0,1] \times [0,1]$.
But, how would I go about showing that the cantor set is homeomoprhic to its product? I'm guessing there's a standard such mapping somewhere out there but I can't seem to find it.
Thanks!

A member of the Cantor set can be written uniquely as $$x = \sum_{i=1}^\infty d_j 3^{-j},\  \text{where all} \ d_j \in \{0,2\}$$  Map this to the pair
$$(y,z) = \left(\sum_{j=1}^\infty d_{2j-1} 3^{-j}, \sum_{j=1}^\infty d_{2j} 3^{-j}\right)$$ 
