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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.

Thanks in advance.

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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.

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