Infinite sets of nonintersecting discs is countable on a plane Prove that any infinite set of non-intersecting discs on the plane is countable. I know that every disk contains rational points and hence there is an objective function from the set of disk to the set of points with rational coordinates. But how would I show that each disk can be mapped to one point with rational coordinates 

If you're concerned about how to pick one point with rational coordinates out of the infinitely many such points in the disk (and you don't want to rely on the Axiom of Choice), take a particular enumeration of the pairs of rationals and choose the first one that is in your disk.
