Inmathematics, aNikodym set is a subset of the unit square in${\displaystyle {\mathbb{R}}^2}$ with complement ofLebesgue measure zero (i.e. with an area of 1), such that, given any point in the set, there is a straight line that only intersects the set at that point.^[1] The existence of a Nikodym set was first proved byOtto Nikodym in 1927. Subsequently, constructions were found of Nikodym sets having continuum many exceptional lines for each point, andKenneth Falconer found analogues in higher dimensions.^[2]

Nikodym sets are closely related toKakeya sets (also known as Besicovitch sets).

The existence of Nikodym sets is sometimes compared with theBanach–Tarski paradox. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.

Mathematicians have also researched Nikodym sets overfinite fields (as opposed to${\displaystyle \mathbb{R}}$).^[3]

• Measure theory
• Paradoxes

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