# $\{0,1\}$-valued continuous functions on the Cantor set

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!

• 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. Nov 10 '18 at 8:43