Finding a one-to-one correspondence from the middle third Cantor set $C$ to the unit sphere $S^2$

Question

I'm familiar with the fact that, if I'm not mistaken, there is a one-to-one correspondence between the unit interval $[0, 1]$ and the unit sphere $S^2$ though I'm not sure explicitly how to find it. My professor told me that in fact there is a one-to-one correspondence between the standard middle third Cantor set $C$ and the unit sphere $S^2$. Is that correct, and if so, how does one construct such a correspondence? This seems to be an interesting idea to investigate further, but I know of no articles that make any mention of this.

Answer 1

It's perhaps a surprising fact that the standard Cantor set has a continuous surjection onto any compact metric space, and in particular the sphere $S^2$. I'm not sure if the map that ends up getting constructed is an injection, but this at least gets you half of the way, continuously! This is discussed in some books on general topology, for instance in Hocking and Young's book Topology . There's an article by Yoav Benyamini that discusses some further consequences of this fact that you can find here.