Nikodym Set

The Nikodym set is a mathematical subset of the unit square that not only has the same area as the unit square, but every point in the set has a straight line that can be drawn through it without intersecting any other points in the set.

Unfortunately, that’s about all the Wikipedia article has to say about it. Where can I find a proof that this set exists? I assume it’s one of those sets that is non-constructible.

It is constructible.

Start on page 215 here.