Geometry puzzle

using the axiom of choice it's fairly easy to construct such a set of open subsets of [math]\mathbb{R}^2[/math]

>what is transfinite induction
You can't just prove something is true of countably infinite circles by proving it for finitely many.

Am I a brainlet because I don't understand this problem? What lines are we talking about?

I assume you had another problem in mind - the one of constructing a set of points such that each line contains exactly two of these points.
I tried something similar but it doesn't really work - taking a maximal set (by Zorn's lemma) of circles such that each line intersects at most, say, 100 of these circles, we may ask if there still could be a line which intersects none of these circles. The problem is that now putting a circle anywhere on this line may indeed could make one line intersect 101 circles...
What I want to say is - there might be no contradiction in assuming that there is a line intersecting no circle in a maximal set of circles.

You might just give a sketch of your idea.