Wacław Sierpiński

How to use this course

Most students meet Wacław Sierpiński through a picture: a triangle with triangles cut out of it, again and again. The picture is real mathematics, but it is a small part of a career of more than 700 papers and 50 books. Sierpiński worked in number theory, set theory, real analysis and topology, and he was the central organizer of the Warsaw school of mathematics between the two world wars. This course follows his life from Russian-ruled Warsaw to the Polish Academy of Sciences, and it works through the main threads of his mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order roughly follows his career: the lattice-point essay (1903–1907), curves and fractals (1912–1916), normal numbers (1916–1917), set theory and descriptive set theory (1914–1947), and number theory again in his last two decades.

Timeline

Warsaw under the tsar, 1882–1906

Wacław Sierpiński was born in Warsaw on 14 March 1882. His father, Konstanty, was a successful physician; his mother was Ludwika, born Łapińska. Warsaw was then the capital of the Congress Kingdom of Poland, a province of the Russian Empire. After the failed rising of 1863 the Russian authorities had pressed hard on Polish culture. Schools and the university taught in Russian, and Polish history and literature were suppressed.

At secondary school a mathematics teacher recognized his talent. In 1899 he entered the Department of Mathematics and Physics of the Imperial University of Warsaw. The decisive teacher there was Georgy Voronoy, a Ukrainian-born number theorist then working on lattices and on the divisor problem. Sierpiński took Voronoy’s methods as his first research tool.

The prize essay

In 1903 the department set a prize question on Voronoy’s contributions to number theory. Sierpiński won the gold medal. The essay contained a new result: the number of lattice points in a disc of radius \(r\) differs from the area \(\pi r^2\) by at most a constant times \(r^{2/3}\). Carl Friedrich Gauss’s elementary bound had been of order \(r\). Lesson 04 proves the stronger estimate.

The essay was accepted for the university’s Russian-language journal. Sierpiński withdrew it rather than publish in Russian, and it appeared in Polish in 1907 in a journal edited by Samuel Dickstein. He graduated in 1904 and taught mathematics and physics at a girls’ school in Warsaw until the school strikes of the 1905 revolution closed it.

He then went to Kraków, in Austrian Galicia, where a Polish university operated freely. At the Jagiellonian University he attended the lectures of the analyst Stanisław Zaremba, along with courses in astronomy and philosophy, and received his doctorate in 1906. The Mathematics Genealogy Project lists both Zaremba and Voronoy as his advisors.

Lwów and the first set theory course, 1906–1914

In 1907 Sierpiński came across a theorem stating that the points of a plane can be specified by a single real coordinate. He wrote to his friend Tadeusz Banachiewicz, later a well-known astronomer, asking how this could be possible. The reply was one word: “Cantor.” Sierpiński read Georg Cantor’s work and changed direction. Set theory became the centre of his research for the next forty years.

In 1908 he was appointed to the University of Lwów, also in Austrian Galicia, where teaching was in Polish. In 1909 he gave there a lecture course devoted entirely to set theory, generally regarded as the first such course anywhere. Cantor’s theory was then barely thirty years old and still controversial. In 1910 he became head of mathematics at Lwów.

His textbooks of this period built a Polish mathematical literature almost from nothing: The Theory of Irrational Numbers (1910), Outline of Set Theory (1912) and The Theory of Numbers (1912). In 1912 he published a new space-filling curve (Lesson 05). His first doctoral students, Stefan Mazurkiewicz and Stanisław Ruziewicz, both completed their degrees at Lwów in 1913. Mazurkiewicz would become his closest collaborator.

In 1914 Sierpiński and Mazurkiewicz published a short note with a startling construction: a set in the plane that splits into two disjoint pieces, each congruent to the whole. The construction needs no axiom of choice. Exercise 8.3 works through it.

Vyatka and Moscow, 1914–1918

When the First World War broke out in the summer of 1914, Sierpiński and his family were in Russian territory. As a professor at an Austrian university he was treated as an enemy subject and interned at Vyatka, a provincial town northeast of Moscow. He kept working. When Dmitri Egorov and Nikolai Luzin in Moscow learned of his situation, they arranged for him to move to the city. He spent the rest of the war there.

The Moscow years were among the most productive of his life. He worked closely with Luzin on functional series, differentiability, the Baire classification of functions, and later on the analytic sets that Mikhail Suslin discovered in 1916–1917 (Lesson 09). Moscow in these years was building its own school of real function theory, and the collaboration shaped both groups.

His own output in these years includes the triangle curve of 1915 and the carpet of 1916 (Lesson 06), both published in the Comptes Rendus of the Paris Academy, and the first explicit absolutely normal number (Lesson 07). He also finished a long survey of the role of the axiom of choice in analysis and set theory, published in 1918.

The Russian revolutions of 1917 and the end of the war changed the map. In 1918 an independent Polish state was proclaimed, and Sierpiński returned first to Lwów and then accepted a post at the University of Warsaw, now a Polish institution. He was 36.

Lattice points in a circle

For the general reader. Draw a large circle on graph paper and count the grid points inside. The count is close to the area of the circle, but not equal to it. How large can the error be? Gauss showed it is at most about the length of the boundary. Sierpiński, at 21, showed it is much smaller, and the problem of finding the true size is still open.

Let \(N(r)=\#\{(m,n)\in\mathbb Z^2 : m^2+n^2\le r^2\}\) and write \(N(r)=\pi r^2+P(r)\). The circle problem asks for the smallest \(\theta\) with \(P(r)=O(r^{\theta+\varepsilon})\) for every \(\varepsilon>0\).

Attach to each lattice point the unit square of which it is the lower-left corner. Squares counted lie inside the disc of radius \(r+\sqrt2\), and the disc of radius \(r-\sqrt2\) is covered by them. So \(\pi(r-\sqrt2)^2\le N(r)\le\pi(r+\sqrt2)^2\), which gives the bound.

Proof sketch (modern form)

Sierpiński used Voronoy’s summation methods. The modern route uses Poisson summation, which is equivalent in substance. Write \(N(r)=\sum_{v\in\mathbb Z^2}\chi(v)\), with \(\chi\) the indicator of the disc. Its Fourier transform is \(\hat\chi(\xi)=r\,J_1(2\pi r|\xi|)/|\xi|\), and the Bessel function satisfies \(|J_1(t)|\lesssim t^{-1/2}\). The sum \(\sum_{\xi\ne0}\hat\chi(\xi)\) does not converge absolutely, so smooth first. Let \(\rho_\delta\) be a bump of total mass 1 supported in a disc of radius \(\delta\), and set \(\chi_\delta=\chi*\rho_\delta\). Then:

1. Monotonicity: \(\sum\chi_{\delta}\) evaluated for radii \(r\pm\delta\) brackets \(N(r)\), at a cost of \(O(r\delta)\) in area.
2. Poisson summation: \(\sum_v\chi_\delta(v)=\pi r^2+\sum_{\xi\ne0}\hat\chi(\xi)\hat\rho(\delta\xi)\), and \(\hat\rho(\delta\xi)\) decays rapidly once \(|\xi|\gg\delta^{-1}\).
3. The tail is therefore bounded by \(\sum_{0<|\xi|\lesssim1/\delta} r^{1/2}|\xi|^{-3/2}\approx r^{1/2}\delta^{-1/2}\).

The total error is \(O(r\delta+r^{1/2}\delta^{-1/2})\). Choosing \(\delta=r^{-1/3}\) balances the two terms and gives \(O(r^{2/3})\).

Where the problem stands

In 1915 G. H. Hardy and Edmund Landau showed independently that \(P(r)\) is not \(O(r^{1/2})\), so \(\theta\ge1/2\). Hardy conjectured that \(\theta=1/2\). Upper bounds have improved slowly, through Johannes van der Corput’s exponential-sum methods to Martin Huxley’s \(\theta\le131/208\approx0.6298\) (2003) and small refinements since. The conjecture remains open.

Curves that fill a square

For the general reader. In 1890 Giuseppe Peano surprised mathematicians with a continuous curve that passes through every point of a square. Such curves showed that the intuitive idea of a curve as a thin, one-dimensional object needed a real definition. Sierpiński added a new example in 1912, a closed curve that fills the square, and his later work helped give curves a precise meaning.

The recursive construction

All such curves are built the same way. Divide the square into \(k\) congruent pieces (4 squares for David Hilbert, 9 for Peano, right isosceles triangles cut along the diagonals for Sierpiński) and the interval into \(k\) equal subintervals. Arrange the pieces in an order where consecutive pieces touch. Repeat inside each piece, matching entry and exit points. The \(n\)-th approximating polygon \(f_n\) satisfies \(\|f_{n+1}-f_n\|_\infty\le C\cdot\lambda^n\) for some \(\lambda<1\), so the \(f_n\) converge uniformly to a continuous limit. Every point of the square lies in a nested sequence of pieces, so it is in the image.

Sierpiński’s curve divides the square along its diagonals into triangles and visits them in a closed loop. Its polygonal approximations are closed paths, which is useful in applications. The region enclosed by the \(n\)-th approximation tends to \(5/12\) of the square.

Why the map cannot be a bijection

A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Removing an interior point disconnects \([0,1]\), while removing any point leaves the square connected. So there is no homeomorphism, and hence no continuous bijection.

The triangle and the carpet

For the general reader. Remove the middle quarter of a triangle, then the middle quarters of the three triangles that remain, and keep going. What survives is a curve with no area that branches at every point. Do the same with a square cut into nine and you get a curve that contains a copy of every curve that can be drawn in the plane. Sierpiński built both in Moscow in 1915 and 1916, as counterexamples. Sixty years later Benoît Mandelbrot made them icons of fractal geometry.

The triangle (1915)

Sierpiński’s paper was titled, in translation, On a curve every point of which is a branch point. Let \(T_0\) be a closed triangle and let \(T_{n+1}\) be obtained from \(T_n\) by removing the open middle triangle (joining the edge midpoints) of each of its \(3^n\) triangles. The triangle is \(S=\bigcap_n T_n\). It is the unique non-empty compact set with

where \(v_1,v_2,v_3\) are the vertices: the attractor of an iterated function system, in the language John Hutchinson introduced in 1981.

The carpet (1916)

Divide a closed square into 9 congruent subsquares and remove the open middle one. Repeat in each of the 8 survivors. The intersection is the Sierpiński carpet \(K\). It is compact, connected, locally connected, has empty interior and Lebesgue measure 0, so it is a curve in the topological sense: a one-dimensional continuum.

So the carpet is a universal plane curve. The proof places a given plane curve inside a square, then uses its one-dimensionality to choose, at each stage, a grid of lines avoiding most of the curve, so that a homeomorphism of the square can push the curve off the removed middle squares. Karl Menger built the three-dimensional analogue, the Menger sponge, in 1926, and proved it universal for curves in any dimension.

This theorem explains why carpets appear throughout geometry and dynamics: as Julia sets of some rational maps, and as limit sets of some Kleinian groups.

Absolutely normal numbers

For the general reader. In the decimal digits of a “typical” number, each digit should appear one time in ten, each pair one time in a hundred, and so on. Émile Borel proved in 1909 that almost every number behaves this way in every base at once. His proof gave no single example. In 1916 Sierpiński wrote one down.

Fix a base \(b\ge2\). For \(x\in[0,1)\) and a word \(w\) of length \(k\) in the digits \(\{0,\dots,b-1\}\), let \(N_b(x,w,n)\) be the number of occurrences of \(w\) among the first \(n\) digits of \(x\). The number \(x\) is normal in base \(b\) if \(N_b(x,w,n)/n\to b^{-k}\) for every \(k\) and every \(w\) of length \(k\). It is absolutely normal if it is normal in every base.

Proof sketch

Fix \(b\) and a digit \(d\). Under Lebesgue measure the digits are independent and uniform. Let \(X_i=\mathbf 1[\text{digit } i=d]-1/b\) and \(S_n=\sum_{i\le n}X_i\). A fourth-moment computation gives \(\mathbb E[S_n^4]\le 3n^2\), so by Markov’s inequality

which is summable in \(n\). By the Borel–Cantelli lemma, almost every \(x\) has \(|S_n|<\varepsilon n\) for all large \(n\). Taking \(\varepsilon=1/m\), \(m=1,2,\dots\), gives simple normality in base \(b\) almost everywhere. Blocks of length \(k\) are handled by splitting positions into \(k\) residue classes, which makes occurrences independent. A countable intersection over bases, words and \(\varepsilon\) still has measure 1.

Sierpiński’s explicit number (1916; published 1917)

Sierpiński made the argument effective. For every base \(b\), digit string \(w\), length \(n\) and tolerance \(1/m\), the numbers whose first \(n\) digits deviate from frequency \(b^{-k}\) by more than \(1/m\) form a finite union of intervals. He chose explicit ranges of \(n\) depending on \(b,m,k\) so that the union \(\Delta\) of all these “bad” intervals has total length less than 1. Every point outside \(\Delta\) is absolutely normal, and the complement of \(\Delta\) in \([0,1]\) is non-empty because of the length bound. Sierpiński’s number is the infimum of that complement, a definite real number. Henri Lebesgue had found a similar construction in 1909, published only in 1917, in the same volume of the Bulletin de la Société Mathématique de France.

Sierpiński’s definition is not obviously computable, because the infimum involves infinitely many conditions. In 2002 Verónica Becher and Santiago Figueira gave a computable variant, and since then fast algorithms have been found. No familiar constant, such as \(\sqrt2\), \(\pi\) or \(e\), is known to be normal in even one base.

Choice, paradoxes and the continuum

For the general reader. Ernst Zermelo’s axiom of choice says that from any family of non-empty sets one can pick one element of each. It sounds harmless, but it implies strange results, and in 1904 many mathematicians refused it. Sierpiński took a different line. He tracked exactly which theorems need the axiom and which follow from its rival, the continuum hypothesis. His 1918 survey and his 1934 book Hypothèse du continu made this a research field.

Bookkeeping for the axiom of choice

In his 1918 survey L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’analyse, Sierpiński listed results in analysis whose known proofs used choice: that a countable union of countable sets is countable, that sequential continuity implies continuity for real functions, that every infinite set has a countable subset, and many more. He asked for each whether choice was really needed. This programme anticipated the independence results of Kurt Gödel (1938) and Paul Cohen (1963), which showed that such questions can have the answer “provably needed.”

A geometric form of CH

The continuum hypothesis (CH) states \(2^{\aleph_0}=\aleph_1\): every infinite set of reals is either countable or in bijection with \(\mathbb R\).

CH ⇒ decomposition. Under CH, well-order \(\mathbb R\) as \(\prec\) in order type \(\omega_1\), so every initial segment is countable. Let \(A=\{(x,y): y\preceq x\}\) and \(B=\{(x,y): x\prec y\}\). The vertical section of \(A\) at \(x\) is \(\{y:y\preceq x\}\), countable; the horizontal section of \(B\) at \(y\) is \(\{x:x\prec y\}\), countable. The converse is Exercise 8.1.

In the same spirit Sierpiński showed in 1924 that CH gives an uncountable set of reals meeting every Lebesgue-null set in a countable set, now called a Sierpiński set. Results of this type filled Hypothèse du continu (1934). Many of them were later shown to be equivalent to CH, and therefore independent of the usual axioms.

GCH implies choice (1947)

Idea. Choice is equivalent to the statement that every set can be well-ordered. For an infinite set \(X\), the Hartogs number \(\aleph(X)\) is the least ordinal that does not inject into \(X\); it exists in ZF and injects into an iterated power set of \(X\). Sierpiński compared, under GCH, the cardinals of \(X,\ \mathcal P(X),\ \mathcal P(\mathcal P(X)),\dots\) with their sums with \(\aleph(X)\). GCH leaves no room for these sums to lie strictly between consecutive power sets, and this forces some iterated power set of \(X\) to be equinumerous with an ordinal. That power set is then well-orderable, and so is \(X\), which injects into it. Adolf Lindenbaum and Alfred Tarski had announced the result in 1926 without proof.

Continua and analytic sets

For the general reader. Topology in the 1910s and 1920s was largely the study of strange subsets of the plane and the line. Sierpiński proved some of its basic structure theorems, and with Luzin he helped found descriptive set theory, which classifies sets of reals by how they can be defined.

Decompositions of continua

A continuum is a compact connected metric space. A connected space is by definition not a disjoint union of two non-empty closed sets. Sierpiński asked about countably many.

Proof sketch. Suppose \(X=\bigsqcup F_n\). Let \(D=X\setminus\bigcup_n\operatorname{int}F_n\), the union of the boundaries. \(D\) is closed, hence compact. Using connectedness and the boundary bumping lemma, one shows \(D\) is non-empty and that each \(F_n\cap D\) is nowhere dense in \(D\). Then \(D=\bigcup_n(F_n\cap D)\) is a non-empty complete metric space written as a countable union of nowhere dense closed sets, contradicting the category theorem of René Baire.

Analytic and projective sets

Henri Lebesgue claimed in 1905 that the projection of a Borel set in the plane is Borel. In Moscow in 1916–1917 Luzin’s student Mikhail Suslin found the error. Projections of Borel sets form a larger class, the analytic sets.

Luzin and Sierpiński developed the theory: analytic sets are Lebesgue measurable, have the property of Baire, and if uncountable contain a perfect subset. In 1925 Luzin and Sierpiński each introduced the projective hierarchy: complements of analytic sets, their projections, the complements of those, and so on. Whether all projective sets are measurable cannot be decided in ZFC, a fact that drives modern descriptive set theory.

Elementary number theory

For the general reader. Sierpiński began and ended with number theory. In his last decades he posed problems that are easy to state and still unsolved: are there infinitely many primes of a given polynomial form, and which numbers \(k\) make \(k\cdot2^n+1\) composite for every \(n\)?

Sierpiński numbers

An odd positive integer \(k\) is a Sierpiński number if \(k\cdot2^n+1\) is composite for every \(n\ge1\).

The method is a covering system, an idea of Paul Erdős (1950). Find primes \(p_1,\dots,p_s\) and residues so that every \(n\) satisfies one congruence \(n\equiv a_i\pmod{\operatorname{ord}_{p_i}2}\), and choose \(k\) by the Chinese remainder theorem so that \(p_i\mid k\cdot2^{n}+1\) whenever \(n\equiv a_i\). Then each \(k\cdot2^n+1\) has a prime factor from the list and, once \(k\) is large, is composite. All \(k\) in an arithmetic progression work.

In 1962 John Selfridge showed that \(78557\) is a Sierpiński number with covering set \(\{3,5,7,13,19,37,73\}\). It is conjectured to be the smallest. The distributed project Seventeen or Bust, now continued by PrimeGrid, has eliminated all smaller candidates except five: 21181, 22699, 24737, 55459 and 67607.

Hypothesis H (Schinzel and Sierpiński, 1958)

Hypothesis H contains the twin prime conjecture (\(x,\ x+2\)), the infinitude of primes \(n^2+1\), and Bunyakovsky’s conjecture. Schinzel and Sierpiński derived many further consequences. Only the case of a single linear polynomial is known: Peter Gustav Lejeune Dirichlet’s theorem on primes in arithmetic progressions.

Sierpiński also conjectured in 1956 that \(5/n=1/x+1/y+1/z\) has a solution in positive integers for every \(n\ge2\), a companion to the Erdős–Straus conjecture for \(4/n\). Both are open.

Warsaw and Fundamenta, 1918–1939

Sierpiński became a professor at the University of Warsaw in 1919 and stayed there for the rest of his career. During the Polish–Soviet War of 1919–1921 he, Mazurkiewicz and Stanisław Leśniewski worked with the cryptologic bureau of the Polish General Staff on breaking Soviet ciphers.

Zygmunt Janiszewski had argued during the war that Polish mathematics should concentrate on a few new fields, set theory, topology and logic among them, and should have its own international journal. With Mazurkiewicz and Sierpiński he planned Fundamenta Mathematicae, devoted to those fields. Janiszewski died in January 1920; the first volume appeared that year. It was among the first specialized mathematics journals, and it published in French, German and English. Sierpiński edited it for decades. Its early volumes carried work by Stefan Banach, Kazimierz Kuratowski, Tarski and many foreign authors.

In 1921 he was elected to the Polish Academy of Learning and became dean of the Warsaw faculty. In 1928 he became chairman of the Polish Mathematical Society and vice-chairman of the Warsaw Scientific Society. He spoke at the International Congress of Mathematicians in Zürich in 1932. He received honorary doctorates from Lwów (1929), San Marcos in Lima (1930) and Sofia (1939), among others.

His doctoral students in Warsaw included Kazimierz Zarankiewicz (1923), Jerzy Neyman (1924), who later founded modern statistical hypothesis testing with Egon Pearson, Adolf Lindenbaum and Zygmunt Zalcwasser (1928), and Edward Marczewski (1932). Sierpiński’s method was to pose many precise problems, often at a seminar or in a short note, and let students attack them. Much of the interwar output of Fundamenta came from problems he had raised.

War, rebuilding and legacy, 1939–1969

The German occupation from 1939 closed Polish universities. Sierpiński worked as a clerk in the Warsaw city council offices and taught in the clandestine underground university. He sent papers to Italy for publication. Many members of the Warsaw and Lwów schools were murdered, among them Ruziewicz (1941), Lindenbaum (1942) and Stanisław Saks. After the Warsaw Uprising of 1944 the Germans burned his house, with his library and correspondence.

After the war he returned to the rebuilt university and to Fundamenta. He received the Stefan Banach Prize of the Polish Mathematical Society in 1946 and the State Scientific Prize, first class, in 1949. In 1947 he published the theorem that the generalized continuum hypothesis implies the axiom of choice (Lesson 08). When the State Institute of Mathematics joined the new Polish Academy of Sciences in 1952, he worked there.

His interest returned to number theory. In 1958 he became editor-in-chief of Acta Arithmetica, and with his student Andrzej Schinzel he formulated Hypothesis H. He retired from his university chair in 1960 but ran a number-theory seminar at the Academy until 1967. Schinzel (1960), Stanisław Świerczkowski (1961) and Andrzej Rotkiewicz (1963) completed doctorates with him in these years. His books Pythagorean Triangles, Cardinal and Ordinal Numbers and 250 Problems in Elementary Number Theory reached a wide international readership. In 1964 he signed the Letter of 34, a protest by Polish intellectuals against censorship.

Foreign honours included membership of the Accademia dei Lincei (1947), the American Academy of Arts and Sciences (1959), honorary membership of the London Mathematical Society (1964) and the Pontifical Academy of Sciences (1967), and an honorary doctorate from Moscow State University (1967). He died in Warsaw on 21 October 1969, aged 87, and is buried in Powązki Cemetery. A lunar crater bears his name.

Legacy

• Set theory: he made the axiom of choice and the continuum hypothesis objects of study, preparing the ground for independence proofs.
• Topology and fractals: the triangle and the carpet, first counterexamples, are now standard models in fractal geometry, dynamics and analysis on fractals.
• Number theory: Sierpiński numbers and Hypothesis H set problems still worked on by both theorists and large computer searches.
• Institutions: Fundamenta Mathematicae and the Warsaw school gave Poland an international place in mathematics within a generation of independence.

Connections

People linked to Wacław Sierpiński across this collection. A name without a link has no page here yet. See Wacław Sierpiński in the network of all 337 lives →

Teachers and advisors

• Georgy Voronoy 1899-1906
• Stanisław Zaremba 1906

Students

• Adolf Lindenbaum 1928
• Andrzej Rotkiewicz 1963
• Andrzej Schinzel 1960
• Antoni Wakulicz 1949
• Edward Marczewski 1932
• Jerzy Neyman 1924
• Kazimierz Zarankiewicz 1923
• Stanisław Ruziewicz 1913
• Stanisław Świerczkowski 1961
• Stefan Mazurkiewicz 1913
• Zygmunt Zalcwasser 1928

Influenced by

• Felix Hausdorff
• Georg Cantor
• Émile Borel

Colleagues and collaborators

• Alfred Tarski
• Henri Lebesgue
• Kazimierz Kuratowski 1920-1969
• Maurice Fréchet
• Stefan Banach
• Dmitri Egorov 1915-1918
• Mikhail Suslin 1916-1918
• Nikolai Luzin 1915-1930s
• Zygmunt Janiszewski

Friends

• Tadeusz Banachiewicz

Also mentioned in this life

• Friedrich Wilhelm Bessel
• Peter Gustav Lejeune Dirichlet
• Paul Erdős
• Leonhard Euler
• Pierre de Fermat
• Joseph Fourier
• Carl Friedrich Gauss
• Kurt Gödel
• G. H. Hardy
• David Hilbert
• Carl Gustav Jacob Jacobi
• Andrey Markov
• Blaise Pascal
• Siméon Denis Poisson

Fields

• set theory 12 lives
• number theory 45 lives
• topology 9 lives
• real analysis 10 lives

On CoursesHub

• Topology in Physics · Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Wacław Sierpiński, MacTutor History of Mathematics.
2. Wacław Sierpiński, Wikipedia.
3. Wacław Franciszek Sierpiński, Mathematics Genealogy Project.
4. K. Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Pergamon, 1980.
5. A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica 21 (1972).

Mathematics

1. W. Sierpiński, Sur une nouvelle courbe continue qui remplit toute une aire plane, Bull. Acad. Sci. Cracovie, Série A (1912).
2. S. Mazurkiewicz and W. Sierpiński, Sur un ensemble superposable avec chacune de ses deux parties, C. R. Acad. Sci. Paris 158 (1914).
3. W. Sierpiński, Sur une courbe dont tout point est un point de ramification, C. R. Acad. Sci. Paris 160 (1915), 302–305.
4. W. Sierpiński, Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée, C. R. Acad. Sci. Paris 162 (1916), 629–632.
5. W. Sierpiński, Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d’un tel nombre, Bull. Soc. Math. France 45 (1917).
6. W. Sierpiński, Hypothèse du continu, Monografie Matematyczne 4, Warsaw, 1934.
7. W. Sierpiński, L’hypothèse généralisée du continu et l’axiome du choix, Fund. Math. 34 (1947).
8. A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958).
9. W. Sierpiński, Cardinal and Ordinal Numbers, 2nd ed., PWN, 1965.
10. G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320–324.
11. V. Becher and S. Figueira, An example of a computable absolutely normal number, Theoretical Computer Science 270 (2002).
12. The \(\Sigma^1_2\) counterparts to statements that are equivalent to the Continuum Hypothesis, arXiv:1201.0382 (survey of Sierpiński’s CH equivalents).
13. Sierpiński number and Sierpiński carpet, Wikipedia.