In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Nikodym set is a subset of the unit square in
with complement of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
zero, such that, given any point in the set, there is a straight line that only intersects the set at that point. The existence of a Nikodym set was first proved by
Otto Nikodym
Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician.
Education and career
Nikodym studied mathematics at the University of Jan Kazimierz (UJK) in Lvov (today's University of Lviv). Imm ...
in 1927. Subsequently, constructions were found of Nikodym sets having continuum many exceptional lines for each point, and
Kenneth Falconer found analogues in higher dimensions.
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.
Mathematicians have also researched Nikodym sets over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s (as opposed to
).
References
{{reflist
Measure theory
Paradoxes