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.
Sierpiński, W. (1924), "Sur l'hypothèse du continu (2ℵ0 = ℵ1)", Fundamenta Mathematicae, 5 (1): 177–187
January 01, 1970
sierpiński, confused, with, sierpiński, space, mathematics, uncountable, subset, real, vector, space, whose, intersection, with, every, measure, zero, countable, existence, independent, axioms, sierpiński, 1924, showed, that, they, exist, continuum, hypothesis. Not to be confused with Sierpinski space In mathematics a Sierpinski set is an uncountable subset of a real vector space whose intersection with every measure zero set is countable The existence of Sierpinski sets is independent of the axioms of ZFC Sierpinski 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 Sierpinski sets are weakly Luzin sets but are not Luzin sets Kunen 2011 p 376 Example of a Sierpinski set editChoose 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 Sa for countable ordinals a For each countable ordinal b choose a real number xb that is not in any of the sets Sa for a lt b 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 xb has only a countable number of elements in each set Sa so is a Sierpinski set It is possible for a Sierpinski set to be a subgroup under addition For this one modifies the construction above by choosing a real number xb that is not in any of the countable number of sets of the form Sa X n for a lt b where n is a positive integer and X is an integral linear combination of the numbers xa for a lt b Then the group generated by these numbers is a Sierpinski set and a group under addition More complicated variations of this construction produce examples of Sierpinski sets that are subfields or real closed subfields of the real numbers References editKunen Kenneth 2011 Set theory Studies in Logic vol 34 London College Publications ISBN 978 1 84890 050 9 MR 2905394 Zbl 1262 03001 Sierpinski W 1924 Sur l hypothese du continu 2ℵ0 ℵ1 Fundamenta Mathematicae 5 1 177 187 Retrieved from https en wikipedia org w index php title Sierpinski set amp oldid 1064626430, wikipedia, wiki, book, books, library,
