VI.77 Wacław Sierpiski

b. Warsaw, 1882; d. Warsaw, 1969

Number theory; set theory; real functions; topology

Sierpiski studied mathematics at the Russian university in Warsaw under the guidance of Georgii Voronoi. In his first paper (1906), he improved GAUSS’S [VI.26] estimate for the difference between the number of lattice points inside the circle x² + y² ≤ N and the area of the circle, showing that it is O (N^1/3).

He became an associate professor at the University of Lwów in 1910, at which point his interest shifted to set theory, on which he wrote a textbook in 1912, only the fifth book ever to be published on the subject. His first important results on set theory were obtained during World War I, which he spent in Russia: in 1915-16 he constructed two curves that were among the first published examples of fractals, one known now as Sierpiski’s gasket and the other as Sierpiski’s carpet. The latter is the set of all points (x, y) in the square [0, 1]² such that, when written out as base 3 decimals, there is no position in which both x and y have a 1. It is also known as Sierpiski’s universal curve, since it contains a homeomorphic image of every planar continuum (a continuum is a compact connected set) without interior points.

In 1917, Souslin had shown that projections of BOREL SETS [III.55] (from the plane into the line, say) need not be Borel. Together with Lusin, Sierpiski proved in 1918 that in fact every analytic set (a projection of a Borel set) is the intersection of ℵ₁ Borel sets (where ℵ₁ is the smallest uncountable cardinal). That same year he also published an important study of THE AXIOM OF CHOICE [III.1] and the role it plays in set theory and analysis, and proved that no continuum can be decomposed into countably many pairwise disjoint nonempty closed subsets.

In 1919 Sierpiski was made a full professor at the new Polish University of Warsaw and in 1920 he founded (together with Janiszewski and Mazurkiewicz) the first specialized mathematical journal, Fundamenta Mathematicae, which was devoted to set theory, topology, and applications. He remained its editor until 1951. Among his results published in volume 1 are a proof that every countable subset of ⁿ without isolated points is homeomorphíc to the rationals; a complete classification of countable compact subsets of ⁿ, obtained jointly with Mazurkiewicz; and a necessary and sufficient condition for a subset of ⁿ to be a continuous image of an interval.

Using the CONTINUUM HYPOTHESIS [IV.22§ 5] (ℵ₁ = 2 ^ℵ₀), he constructed an UNCOUNTABLE SET [III.11] of reals, now known as a Sierpiski set, such that every uncountable subset of it is nonmeasurable (1924); and also a one-to-one mapping of the line into itself that maps sets of MEASURE ZERO [III.55] to sets of first category, in such a way that every set of the first category is obtained (1934). The former result is highly paradoxical (there is no explicit example of a nonmeasurable set); the latter has led, thanks to Erds, to the following duality principle. Let P be any proposition involving solely the notions of measure zero, first category, and pure set theory. Let P* be the proposition obtained from P by interchanging the terms “set of measure zero” and “set of first category.” Then P and P* are equivalent, assuming the continuum hypothesis.

Sierpiski wrote a monograph devoted to the continuum hypothesis in 1934, entitled Hypothèse du Continu. Together with TARSKI [VI.87] he introduced the notion of STRONGLY INACCESSIBLE CARDINALS [IV.22 §6] (1930), meaning cardinals m that cannot be obtained as products of fewer than m cardinals less than m. He also worked in Ramsey theory, giving a limitation on infinite extensions to Ramsey’s theorem. To be precise, Ramsey had proved that, whenever one finitely colors the pairs from the natural numbers, there is an infinite monochromatic subset (i.e., a subset all of whose pairs have the same color); Sierpiski showed that, by contrast, one can 2-color the pairs from a ground set of size ℵ₁ in such a way that there is no monochromatic subset of size ℵ₁. He also deduced the axiom of choice from the generalized continuum hypothesis (formulated without cardinals in 1947).

In his old age he returned to number theory and became the editor of Acta Arithmetica (1958–69).

Further Reading

Sierpiski, W. 1974–76. Oeuvres Choisies. Warsaw: Polish Scientific.

Andrzej Schinzel