I'm working with this article of Stevo Todorcevic and I am stuck with the fact that not all orthogonal definable families can be separated. More concretely, I am unable to prove that the sets $A=\{a_i\}$ and $B$ are Borel sets (taking the topology of the Cantor set). I've managed to notice that $$A=\bigcup_{n,m=1}^{\infty} {\{2^n(2m+1)}\}$$ and I was trying to prove that these singletons are open sets. However, I think there's something that I'm missing or simply I've gone the wrong way.
I would appreciate any help.
Each set $a_i$ is simply a point in the Cantor set $C$, so $A$ is countable and therefore Borel. Suppose that $b\in C$ and $b\perp a_i$ for each $i$. Each $n\in b$ can be written uniquely as $2^{k(n)}m(n)$, where $m(n)$ is odd. For each $i$ let $b_i=\{n\in b:k(n)=i\}$; then each $b_i$ is finite. Indeed, $B$ is the set of all $b\in C$ such that $b_i$ is finite for each $i$. For a fixed $i$, the set of $b\in C$ such that $b_i$ is finite is $F_\sigma$, so $B$ is $F_{\sigma\delta}$.