suppose i define $$C=\{ (x,y)\in \mathbb{R}^2 | x^2+y^2=1\}$$ $$A=\{ (x,y)\in \mathbb{R}^2 | x,y\in \mathbb{Q} \wedge x^2+y^2<1\}$$
and i define a graph $G= \langle C,E \rangle$
when $E$ is the set edges, and 2 vertices are connected iff the chord between them does not contain any points in $A$ (every point on the chord has at least one irrational coordinate).
i want to prove that for every countable set $B\subseteq C$ there is a point $(x,y)\in C-B $ that has an edge to any point in $B$.
It feels like transfinite induction but im not really sure on how to start it
thank you!
