Continuity of a mapping $C\to C^2$, $C$ being the Cantor set

Question

I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$.

Now we consider the function $$f:C\to C^2 \\ 0.a_1a_2a_3... \mapsto \left( 0.a_1a_3a_5... , 0.a_2a_4a_6... \right)$$ I would like to show that $f$ is continuous.

I appreciate any advice/hints.

Answer 1

Let $f=(f_1,f_2)$, with $f_1,f_2: C\to C$.

Clearly, if $\lvert x-y\rvert<\delta$, and $\delta<\dfrac{2}{3^{2n+1}}$, for some $n$, then the first $2n+1$ digits in the ternary expansions of $x$ and $y$ agree, and hence so do the first $n$ ternary digits of $f_i(x)$ and $f_i(y)$, $i=1,2$, and thus $$ \lvert\, f_1(x)-f_1(y)\rvert,\,\lvert\, f_2(x)-f_2(y)\rvert<\dfrac{2}{3^{n}}. $$ Hence, for every $\varepsilon>0$, there exists a $\delta=\dfrac{2}{3^{k}}$, where $$ k=2\lfloor\log_3 (\varepsilon/2)\rfloor+1, $$ such that...

Answer 2

Let $h$ be the mapping between the so-called Cantor cube $K = \{0,1\}^{\mathbb{N}}$ (in the product topology, where $\{0,1\}$ is discrete) and the Cantor set (as a subset of $[0,1]$) that sends a sequence $(x_i)_{i \ge 1} \in K$ to the point in $C$ with ternary expansion $0.a_1a_2\ldots$ where $a_i = 2x_i$ for all $i$.

Key to the proof is the fact that $h$ is a homeomorphism. Once you know that, your $f$ is just the index shuffling homeomorphism between $K$ and $K \times K$, which is easily shown to be continuous using the universal property of maps into product (continuous iff all the compositions with projections are).