Subset of the plane that intersects every line exactly twice

This theorem is originally to S. Mazurkiewicz. In Sierpinski's book "Cardinal and Ordinal Numbers", the theorem is proved on page 449 (Chapter 17, Section 2). In fact the following more general result by F. Bagemihl ("A theorem on intersections of prescribed cardinality", Annals of Math., 55 (1952), p. 34, jstor) holds:

Theorem: Suppose that every straight line $S$ lying in a plane is associated with cardinal number $m_S$ such that $2 \leq m_S \leq 2^{\aleph_0}$. Then there exists a plane set $Q$ such that the cardinality of $Q \cap S$ is $m_S$ for every straight line $S$ in the plane.

(I am mentioning Sierpinski's book because it might be more easily available than 1952 issues of Annals of Mathematics.)

By AC, choose a cardinal well-ordering of the lines in in the plane and any well-ordering of all the points.

We proceed by transfinite induction.

Suppose $A_l$ is a set of points, no three colinear, and let $B_l$ be the set of lines spanned by points of $A_l$, and let $C_l=\cup B_l$. Suppose further that $l'\prec l$ implies $l'\in B_l$. Note that $|A_l| \leq |B_l| < |l|$, so that $$|C_l\cap l| = |\cup_{l'\in B_l} l\cap l'| < |l| .$$

• If $l\in B_l$, let $A_{Sl} = A_l$.
• If $a\in l \cap A_l$, take the minimal point $b\in l\backslash C_l$ and let $A_{Sl} = A_l\cup \{b\} $.
• If $A_l\cap l = \emptyset$, take the minimal two points $a,b\in l\backslash C_l$, and let $A_{Sl}=A_l\cup\{a,b\}$.
• Otherwise if $l'\ $ is a limit ordinal, let $A_{l'} = \bigcup_{l\prec l'} A_l$.

It is easy to check that $A_{l'}$ has no three points colinear --- they'd have to all be in some $A_l$ for $l\prec l'$. The final union $\bigcup_l A_l$ has exactly two points on each line $l$.

It's still an open problem whether there exists a Borel set that is a 2-point set (or $n$-point set for that matter). There is a PhD-thesis from my alma mater (Free University Amsterdam) by K.Bouhjar (under Jan van Mill) (2002) called "On the structure of N-point sets". This has some more info on sets like this. One result is that no $n$-point set in the plane can be $\sigma$-compact.