(Remark. A set that intersects every line of the plane in exactly n points is called an n-set. A 1-set is clearly impossible; this question asks whether 2-sets exist.)

Hint 1

The answer is Yes, although it's hard to visualize such a set. This result was proved by Stefan Mazurkiewicz in 1914, and we will show his proof. In this article we will show how to construct the set using transfinite recursion. Surprisingly the construction uses almost no facts about geometry or the plane, but it does use some facts about transfinite numbers, so you should be familiar with those. (Most books on set theory discuss this topic.)

Let's practice on a simpler version of the problem before we take a leap into the transfinite. Suppose we are given a finite collection of straight lines in the plane, L₁, ..., L_N. Describe a method for generating a set of points in the plane such that (1) each line of this collection intersects exactly two members of the set, and (2) any line in the plane intersects at most two members of the set. An easy way to do this would be to draw a very large circle, large enough that every line intersects it; but this method does not generalize well, so think of some other method where you select the points one at a time.

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