Sierpiński set

In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. Sierpiński (1924) showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ℵ1 is true. Sierpiński sets are weakly Luzin sets but are not Luzin sets (Kunen 2011, p. 376).

Not to be confused with Sierpiński space.

Example of a Sierpiński set

Choose a collection of 2ℵ0 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 Sα for countable ordinals α. For each countable ordinal β choose a real number xβ that is not in any of the sets Sα 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 xβ has only a countable number of elements in each set Sα, so is a Sierpiński set.

It is possible for a Sierpiński set to be a subgroup under addition. For this one modifies the construction above by choosing a real number xβ that is not in any of the countable number of sets of the form (Sα + X)/n for α < β, where n is a positive integer and X is an integral linear combination of the numbers xα 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
  • Kunen, Kenneth (2011), Set theory, Studies in Logic, 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001
  • Sierpiński, W. (1924), "Sur l'hypothèse du continu (2ℵ0 = ℵ1)", Fundamenta Mathematicae, 5 (1): 177–187
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