- the area (i.e. two-dimensional Lebesgue measure) of N is 1;
- for every point x of N, there is a straight line through x that meets N only at x.
Analogous sets also exist in higher dimensions.
The existence of such a set as N was first proved in 1927, by Polish mathematician Otto M. Nikodym. The existence of higher-dimensional Nikodym sets was first proved in 1986, by British mathematician Kenneth Falconer.
Nikodym sets are closely related to Kakeya sets (also known as Besicovitch sets).
The existence of Nikodym sets is sometimes compared with the Banach–Tarski paradox. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.