Constructive proof of representation for open balls in Cantor space

Ok, let $B$ be an open ball in $\mathfrak{C}$. For $x\in\mathfrak{C}$, let $U_{x,n}\subseteq\mathfrak{C}$ denote the set of those elements in the Cantor set whose first $n$ digits in their base 3 representation agree with those of $x$. I surmise you've already shown that, for each $x\in B$, we can choose a large enough $n_x\gg0$, such that $U_{x,n_x}\subseteq B$. Then, I claim that $B=\bigcup_{x\in B}U_{x,n_x}$. This is the desired representation. To verify the claim, note that if $y\in B$, then $y\in U_{y,n_y}\subseteq\bigcup_{x\in B}U_{x,n_x}$. This proves one inclusion. On the other hand, $U_{x,n_x}\subseteq B$ for each $x\in B$ by construction, whence $\bigcup_{x\in B}U_{x,n_x}\subseteq B$, which is the other inclusion.