In mathematics, a Sierpiński set is an

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...

subset of a

real vector space Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. showed that they exist if the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

is true. On the other hand, they do not exist if

Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...

for ℵ₁ is true. Sierpiński sets are weakly Luzin sets but are not

Luzin set In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: t ...

s .

Example of a Sierpiński set

Choose a collection of 2^ℵ₀ measure-0 subsets of R such that every measure-0 subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as ''S''_''α'' for countable ordinals ''α''. For each countable ordinal ''β'' choose a real number ''x''_''β'' that is not in any of the sets ''S''_''α'' for ''α'' < ''β'', which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set ''X'' of all these real numbers ''x''_''β'' has only a countable number of elements in each set ''S''_''α'', so is a Sierpiński set. It is possible for a Sierpiński set to be a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

under addition. For this one modifies the construction above by choosing a real number ''x''_''β'' that is not in any of the countable number of sets of the form (''S''_''α'' + ''X'')/''n'' for ''α'' < ''β'', where ''n'' is a positive integer and ''X'' is an integral linear combination of the numbers ''x''_''α'' for ''α'' < ''β''. Then the group generated by these numbers is a Sierpiński set and a group under addition. More complicated variations of this construction produce examples of Sierpiński sets that are subfields or real-closed subfields of the real numbers.

References

* * {{DEFAULTSORT:Sierpinski set Measure theory Set theory