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Let $P$ be the ternary Cantor set. I need to determine all continuous functions $f:P\to \mathbb{F}_2$ where $\mathbb{F}_2=\{0,1\}$ is the finite field with two elements.

We know that $P$ is homeomorphic to the infinite product $\prod_{i=0}^\infty\mathbb{F}_2$. The projection $\pi_j$ to the $j$th component is of course a desired one. Also if we let the point-wise addition and production of functions $f,g:P\to\mathbb{F}_2$ by $(f+g)(x)=f(x)+g(x)$ and $(fg )(x)=f(x)g(x)$, then any product and addition of projections is also a continuous map.

It seems to me that all elements of $C(P,\mathbb{F}_2)$ are finite sums of production of projections, but I stuck!

Thanks for your helps.

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  • $\begingroup$ Fix the typo in the third line. It's actually two. $\endgroup$
    – Avinash N
    Nov 10 '18 at 7:58
  • $\begingroup$ @AvinashN Done. Thanks. $\endgroup$
    – Qurultay
    Nov 10 '18 at 7:59
  • $\begingroup$ Also the "Contor" set in the first sentence should be a "Cantor" set $\endgroup$
    – celtschk
    Nov 10 '18 at 8:02
  • $\begingroup$ @celtschk Well by "the" Cantor set, I mean the classical Cantor set, which obtained by removing middle third interval in each step. $\endgroup$
    – Qurultay
    Nov 10 '18 at 8:06
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    $\begingroup$ Hint: the open subgroups of $P=\mathbb{F}_2^{\mathbb{N}}$ are the finite-index subgroups. So we know what the clopen sets look like, and hence what continuous functions there are. $\endgroup$ Nov 10 '18 at 8:43

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