# Explicit homeomorphism between product of Cantor sets onto the Cantor set

I want to find an explicit homeomorphism $$\varphi: C\times C \longrightarrow C$$ where $$C$$ denotes the Cantor set. The hint is to use the base $$3$$ expansion of the elements of the Cantor set. My two guesses are:

• $$\varphi: C \times C \longrightarrow C$$ given by $$\left(\sum \limits_{n=1}^\infty \frac{a_n}{3^n},\sum \limits_{n=1}^\infty \frac{b_n}{3^n} \right) \mapsto \sum \limits_{k=0}^\infty \frac{a_{k+1}}{3^{2k+1}} + \sum \limits_{k=1}^\infty \frac{b_{k}}{3^{2k}}$$.
I was trying to prove that the latter satisfies the Lipschitz condition where $$C \times C$$ is equipped with taxicab metric inherited from $$\mathbb{R}^2$$.

• $$\varphi: C \times C \longrightarrow C$$ given by $$\left(\sum \limits_{n=1}^\infty \frac{a_n}{3^n},\sum \limits_{n=1}^\infty \frac{b_n}{3^n} \right) \mapsto \sum \limits_{n=1}^\infty \frac{|a_n-b_n|}{3^n}$$ It is at least well-defined (?) but I have no clue how to prove the continuity neatly.

Your idea is correct, but not properly developped. See my answer to Space filling curve. Fact 1 is nothing else than that you have a continuos bijection $$\varphi : P = \prod_{n=1}^\infty \{0, 1 \} \to C, \varphi((x_n)) = \sum_{n=1}^\infty \frac{2x_n}{3^n}$$. Here $$P$$ is the infinite product of discrete spaces $$\{0,1\}$$ which are compact, whence $$P$$ is compact and $$\varphi$$ is a homeomorphism. Fact 3 is trivial, it says nothing else than that permuting coordinates in a product yields a homeomorphism.