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.