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).

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 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

Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001
Sierpiński, W. (1924), "Sur l'hypothèse du continu (2^ℵ₀ = ℵ₁)", Fundamenta Mathematicae, 5 (1): 177–187, doi:10.4064/fm-5-1-177-187