Felix Hausdorff

How to use this course

Every student of topology meets Felix Hausdorff in the first week, through the separation axiom that bears his name. Students of fractals meet him through Hausdorff dimension, students of Lie theory through the Baker–Campbell–Hausdorff formula, and set theorists through gaps, cofinality and the maximal principle. Fewer know that he began as an astronomer, published philosophy and a successful stage comedy under a pen name, and died with his wife and her sister rather than be deported by the Nazi regime. This course tells both stories.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order of his work, from ordered sets (1906–09) through the 1914 book to measure and analysis (1916–23); the Lie-theoretic formula of 1906 closes Part II because it stands somewhat apart.

Timeline

Breslau and Leipzig, 1868–1891

Felix Hausdorff was born on 8 November 1868 in Breslau, in Prussian Silesia, now Wrocław in Poland. His father, Louis Hausdorff, was a Jewish textile merchant; his mother was Hedwig Tietz. When Felix was a small boy the family moved to Leipzig, where he grew up in a prosperous household.

From 1878 to 1887 he attended the Old St Nicholas School in Leipzig, a classical gymnasium, and left with the highest grade. His interests were wide: music, literature and philosophy as much as science. He chose to study mathematics and astronomy at the University of Leipzig, with a summer semester at Freiburg in 1888 and a winter semester at Berlin in 1888–89.

The astronomer

At Leipzig his teachers were the astronomer and mathematician Heinrich Bruns and the analyst Adolph Mayer. His dissertation of 1891, Zur Theorie der astronomischen Strahlenbrechung, treated the refraction of starlight in the atmosphere, a problem of applied analysis that required careful modelling of the air’s layered density. Further papers on refraction and the absorption of light in the atmosphere followed, and his habilitation of 1895 at Leipzig continued in the same direction. Nothing in this early work announced the founder of general topology.

Paul Mongré, 1891–1904

As a Privatdozent at Leipzig from 1895, Hausdorff taught mathematics while leading a second life as a writer. Under the pen name Paul Mongré he published Sant’ Ilario: Gedanken aus der Landschaft Zarathustras (1897), a book of aphorisms written in the spirit of Friedrich Nietzsche, whose work he admired. A year later came Das Chaos in kosmischer Auslese (1898), an epistemological essay that argued against metaphysical claims about the world as it is in itself. It used set-theoretic ideas, including Georg Cantor’s infinite sets, to show how many different worlds would be compatible with the same experience.

A volume of poems, Ekstases, followed in 1900. His one-act comedy Der Arzt seiner Ehre (The Doctor of His Honour), a satire on duelling, was published in 1904 and staged with success from 1912. He also wrote essays for literary magazines.

In 1899 he married Charlotte Goldschmidt; their daughter Lenore, was born in 1900. The philosophical work mattered for the mathematics. Thinking about the structure of time and space in Das Chaos led him to Cantor’s set theory, which he began to study seriously around 1900.

Turning to sets, 1901–1913

In the summer of 1901 Hausdorff lectured at Leipzig on set theory, at a time when very few universities offered such a course. In December 1901 he was appointed associate professor (an außerplanmäßiger Extraordinarius, without the full rights of a chair). His mathematical output now changed character entirely.

Between 1906 and 1909 he published a series of long papers on ordered sets: the classification of order types, the notion of cofinality, the sets he called \(\eta_\alpha\)-sets, and the construction of what is now called a Hausdorff gap. Lesson 04 presents them. In 1906 he also published a paper on the symbolic exponential formula in group theory, the subject of Lesson 10. Cantor had left order theory largely at the level of countable types; Hausdorff created its general theory, and some of his questions were settled only with the methods of modern set theory.

In 1910 he moved to the University of Bonn, encouraged by the geometer Eduard Study, and in 1913 he became a full professor at Greifswald, a small university on the Baltic coast. There he completed the book that made his reputation.

Ordered sets

For the general reader. The rational numbers and the real numbers are both lists with no gaps at the ends and something between any two members, yet they are very different in size. Hausdorff asked what kinds of ordered lists can exist at all. He found ways to measure how long a list must be to reach its end, and he found a gap in a natural ordering that no single function can fill.

A linear order is dense if between any two elements there is a third. Cantor proved the basic result.

Back and forth. Enumerate \(A=\{a_0,a_1,\dots\}\) and \(B=\{b_0,b_1,\dots\}\). Build finite order-preserving maps \(f_0\subseteq f_1\subseteq\cdots\). At even steps, add the first \(a_i\) not in the domain; density and the absence of endpoints in \(B\) give an image in the right interval. At odd steps, add the first \(b_j\) not in the range, symmetrically. The union is an isomorphism.

Hausdorff’s generalization

For an ordinal \(\alpha\), Hausdorff called a linear order an \(\eta_\alpha\)-set if for any two subsets \(X<Y\) of size less than \(\aleph_\alpha\) (every element of \(X\) below every element of \(Y\)) there is an element strictly between them. \(\mathbb Q\) is the \(\eta_0\)-set. He proved that \(\eta_\alpha\)-sets are universal (every order of size \(\aleph_\alpha\) embeds in one) and, under the generalized continuum hypothesis, unique in the appropriate size. These are the prototypes of saturated models in model theory.

Cofinality

The cofinality \(\operatorname{cf}(\alpha)\) of a limit ordinal is the least order type of a cofinal (unbounded) subset. Hausdorff introduced the notion and distinguished regular cardinals (\(\operatorname{cf}\kappa=\kappa\)) from singular ones. He asked whether there could be regular limit cardinals, which are the weakly inaccessible cardinals at the root of large-cardinal theory.

The principle is equivalent, over the other axioms of set theory, to the axiom of choice and to Zorn’s lemma (1935), which it anticipated.

Gaps

For \(f,g:\mathbb N\to\mathbb N\), write \(f<^*g\) if \(f(n)<g(n)\) for all but finitely many \(n\). A pair of sequences \((f_\alpha)_{\alpha<\omega_1}\) increasing and \((g_\beta)_{\beta<\omega_1}\) decreasing, with every \(f_\alpha<^*g_\beta\), is a gap if no \(h\) satisfies \(f_\alpha<^*h<^*g_\beta\) for all \(\alpha,\beta\).

By contrast, a countable increasing sequence below a countable decreasing one can always be separated (Exercise 4.2). The gap shows that the ordering cannot be “completed” in a simple way, and it remains a basic tool in set-theoretic topology and in the theory of forcing axioms.

Topological and metric spaces

For the general reader. Before 1914 each kind of space had its own theory of limits and continuity: the line, the plane, spaces of functions. Hausdorff wrote down a short list of rules about “neighbourhoods” from which the common theory follows. With that list, and the related notion of a metric space, topology became a subject in its own right.

The Grundzüge der Mengenlehre appeared in April 1914, dedicated to Cantor. Its first half is a superb account of cardinal and ordinal arithmetic and ordered sets; its second half founds general topology. Maurice Fréchet had introduced abstract metric spaces in 1906, and others had studied limit structures; Hausdorff chose neighbourhoods as primitive and gave the theory its lasting form.

Axioms (A)–(C) say that the neighbourhoods form a base of open sets for a topology. Axiom (D) is the separation axiom now called Hausdorff or \(T_2\); later authors made it optional. In the book Hausdorff then developed closed sets, closure, interior, connectedness, compactness (for metric spaces), completeness and the completion of a metric space, separability and the Borel hierarchy.

The Hausdorff distance

For non-empty bounded sets \(A,B\) in a metric space,

The Hausdorff metric is the natural setting for fractals defined as fixed points of contractions (Lesson 08) and for Gromov–Hausdorff convergence in geometry.

The size of Borel sets

For the general reader. Cantor conjectured that every infinite set of real numbers is either countable or as large as the whole line. Nobody could prove it for arbitrary sets, and in fact it cannot be decided from the usual axioms. In 1916 Hausdorff and Pavel Alexandrov independently proved it for the very large family of Borel sets, which contains every set an analyst normally meets.

The Borel sets of a metric space are the smallest family containing the open sets and closed under complements and countable unions. A non-empty closed set is perfect if it has no isolated points. A perfect subset of \(\mathbb R\) contains a copy of the Cantor set and therefore has cardinality \(2^{\aleph_0}\).

The closed case goes back to Cantor and Ivar Bendixson: every closed \(F\subseteq\mathbb R\) is the disjoint union of a perfect set and a countable set. The Borel case needs a new idea. Alexandrov’s and Hausdorff’s methods were developed by Nikolai Luzin and Mikhail Suslin into the theory of analytic sets, the beginning of descriptive set theory. Hausdorff also introduced the notation \(G_\delta\) and \(F_\sigma\) for countable intersections of open sets and countable unions of closed sets, still in universal use.

The Hausdorff paradox

For the general reader. Can one assign an “area” to every subset of a sphere, in a way that adds up correctly and does not change when the set is rotated? Hausdorff showed that the answer is no. He cut the sphere, minus a few points, into three pieces so strange that one piece is congruent to each of the others and also to the union of the other two. Ten years later this led to the Banach–Tarski paradox.

Proof sketch

Hausdorff found rotations \(\varphi\) of order 2 and \(\psi\) of order 3 about axes at a suitable angle such that the group they generate is the free product \(G=\mathbb Z/2*\mathbb Z/3\): every element is a unique alternating word in \(\varphi\) and \(\psi^{\pm1}\). Each non-identity rotation fixes two points; let \(D\) be the countable set of all these fixed points. Then \(G\) acts freely on \(S^2\setminus D\).

Partition \(G\) itself into \(A_0,B_0,C_0\) with \(\varphi A_0=B_0\cup C_0\), \(\psi A_0=B_0\), \(\psi^2A_0=C_0\). Hausdorff built this partition by induction on the length of reduced words, using the uniqueness of normal forms. Using the axiom of choice, pick one point from each \(G\)-orbit in \(S^2\setminus D\) to form a set \(M\), and put \(A=A_0M\), \(B=B_0M\), \(C=C_0M\). Then \(\varphi A=B\cup C\), \(\psi A=B\) and \(\psi^2A=C\).

If \(\mu\) were a finitely additive invariant measure with \(\mu(D)=0\), then \(\mu(A)=\mu(B)=\mu(C)\) and \(\mu(A)=\mu(B)+\mu(C)=2\mu(A)\), so all vanish and \(\mu(S^2)=0\). Removing the hypothesis on \(D\) takes one more step. In 1924 Stefan Banach and Alfred Tarski strengthened the result: a solid ball can be cut into finitely many pieces and reassembled into two balls of the same size. In the plane and on the line, by contrast, Banach showed that invariant finitely additive measures do exist; the difference lies in the free subgroups of \(\mathrm{SO}(3)\), an observation that led John von Neumann to the notion of amenable groups.

Hausdorff measure and dimension

For the general reader. A line has dimension 1 and a square dimension 2. Hausdorff found a way to measure the dimension of any set, and discovered that many sets have dimensions that are not whole numbers. The Cantor dust has dimension about 0.63; a coastline or a snowflake curve has dimension between 1 and 2. Sixty years later Benoît Mandelbrot called such sets fractals.

Constantin Carathéodory had in 1914 constructed linear (one-dimensional) measure in \(\mathbb R^n\) by covering sets with small pieces. In Dimension und äußeres Maß, published in Mathematische Annalen in 1919, Hausdorff replaced the length of the pieces by an arbitrary power of their diameter.

Why the two descriptions agree. If \(t>s\) and \(\operatorname{diam}U_i<\delta\), then \(\sum(\operatorname{diam}U_i)^t\le\delta^{t-s}\sum(\operatorname{diam}U_i)^s\). Hence \(\mathcal H^t_\delta(E)\le\delta^{t-s}\mathcal H^s_\delta(E)\), and if \(\mathcal H^s(E)<\infty\) then \(\mathcal H^t(E)=0\). So \(s\mapsto\mathcal H^s(E)\) is \(\infty\) below one critical value and 0 above it. On \(\mathbb R^n\), \(\mathcal H^n\) is a constant multiple of Lebesgue measure.

Hausdorff computed in 1919 that the middle-thirds Cantor set has dimension \(\log2/\log3\). The general theorem above is due to Patrick Moran (1946). The upper bound is easy: covering by the \(m^k\) pieces at level \(k\) gives \(\mathcal H^s_\delta\le\sum r_{i_1}^s\cdots r_{i_k}^s=1\). The lower bound requires a mass-distribution argument. Abram Besicovitch developed the geometric theory of Hausdorff measure from the late 1920s, so that the dimension is sometimes called the Hausdorff–Besicovitch dimension.

Moments, summation, Fourier series

For the general reader. If you know all the averages \(\int x^n\,d\mu\) of a distribution of mass on an interval, can you recover the distribution, and which lists of numbers can arise this way? Hausdorff answered both questions for the interval \([0,1]\). In the same years he found the sharp inequality that relates the size of a function to the size of its Fourier coefficients.

For a sequence \((\mu_n)_{n\ge0}\) define the difference operator \((\Delta\mu)_n=\mu_{n+1}-\mu_n\). The sequence is completely monotone if \((-1)^k(\Delta^k\mu)_n\ge0\) for all \(k,n\ge0\).

Necessity is a one-line computation: \((-1)^k(\Delta^k\mu)_n=\int_0^1x^n(1-x)^k\,d\sigma\ge0\). Sufficiency builds discrete measures putting mass \(\binom Nk(-1)^{N-k}(\Delta^{N-k}\mu)_k\ge0\) at \(k/N\), and takes a weak limit; Bernstein polynomials make the convergence work. Uniqueness follows from the Weierstrass approximation theorem.

Hausdorff’s motive was summability. A lower-triangular matrix method transforms partial sums \(s_k\) into means \(t_n=\sum_k\binom nk(-1)^{n-k}(\Delta^{n-k}\mu)_k\,s_k\). He showed that such a Hausdorff method is regular (it sums every convergent series to its sum) exactly when \(\mu\) is the moment sequence of a signed measure of bounded variation with no atom at 0 and total mass 1. The Cesàro and Hölder means are special cases.

William Henry Young had proved the case where \(q\) is an even integer in 1912; Hausdorff extended it to all \(q\ge2\). The endpoints are Parseval (\(p=2\)) and the trivial bound \(|\hat f(n)|\le\|f\|_1\) (\(p=1\)), and today the inequality is the standard first example of the Riesz–Thorin interpolation theorem.

The exponential formula for Lie groups

For the general reader. For ordinary numbers \(e^xe^y=e^{x+y}\). For matrices this fails, because the order of multiplication matters. Hausdorff proved that the correction can always be written using only commutators \(XY-YX\). This is why a Lie group is determined near the identity by its Lie algebra.

John Campbell and Henry Baker had made the claim and given partial arguments; Hausdorff’s paper Die symbolische Exponentialformel in der Gruppentheorie (1906) gave the first complete proof, by deriving a differential equation for \(Z\) along the path \(e^Xe^{tY}\) and showing that its solution stays in the Lie algebra. Eugene Dynkin found an explicit formula for all terms in 1947. Since the series converges for small matrices, it shows that the group law near the identity is determined by the bracket, which is the core of Lie’s third theorem.

Greifswald and Bonn, 1913–1935

Hausdorff was in Greifswald from the summer of 1913 to 1921. The Grundzüge appeared in April 1914, months before the war, and its influence grew slowly but steadily; the Polish school of Wacław Sierpiński and Kazimierz Kuratowski and the Moscow school of Alexandrov and Pavel Urysohn built directly on it. In Greifswald he also wrote the papers on Borel sets (1916) and on dimension (1919).

In 1921 he returned to Bonn as full professor, where he taught until 1935. His colleagues included Eduard Study and later Otto Toeplitz, and the Bonn mathematics of the 1920s was among the liveliest in Germany. His lectures were admired for their clarity and finish. He had relatively few doctoral students, among them Karl Bögel (1924), Gustav Steinbach (1930) and Franz Hallenbach (1933). The paper on moments and summation (1921) and the Hausdorff–Young inequality (1923) date from these years, as does the second edition of his book, Mengenlehre (1927), much rewritten and with an emphasis on descriptive set theory. The new edition cut much of the general topology, so the 1914 edition remains the classic source for it.

After 1904 he largely gave up literary writing. The success of his comedy in 1912 and after was a late echo of the Paul Mongré years.

Persecution, death and legacy

As a Jew, Hausdorff was exposed to the Nazi laws from 1933. He kept his post for two more years as a long-serving civil servant, but after some back-and-forth he was made emeritus on 31 March 1935. He continued to work privately on set theory and topology, leaving extensive manuscripts. After the November pogrom of 1938 the family’s situation worsened. In 1939 he asked Richard Courant, then in New York, about a research fellowship that might allow him to emigrate; nothing came of it. His daughter Lenore survived the war and died in Bonn in 1991.

From the autumn of 1941 the Hausdorffs, like all German Jews, were forced to wear the yellow star. In January 1942 Felix and Charlotte Hausdorff and her sister Edith Pappenheim were ordered to move to the internment camp at Endenich, a former monastery in Bonn from which Jews were deported to the east. On the night of 25 January 1942 the three took an overdose of barbiturates. Felix and Charlotte died by the next morning; Edith Pappenheim died a few days later without regaining consciousness. In a farewell letter to his friend, the lawyer Hans Wollstein, Hausdorff wrote that they had solved the problem “in a different manner”. Wollstein was later murdered at Auschwitz. Among the few who had helped the family to the end was the Bonn mathematician Erich Bessel-Hagen.

Hausdorff is buried in the Poppelsdorf cemetery in Bonn. Before his death he entrusted his papers to the Egyptologist Hans Bonnet, who preserved them through the war; they are now in the University and State Library of Bonn.

Legacy

• Topology: the axiomatic definition of topological space, the Hausdorff separation axiom, the Hausdorff metric on closed sets.
• Set theory: cofinality, regular and singular cardinals, \(\eta_\alpha\)-sets, Hausdorff gaps, the maximal principle, the \(G_\delta\)/\(F_\sigma\) notation, the cardinality of Borel sets.
• Measure and geometry: Hausdorff measure and dimension, the language of fractal geometry and geometric measure theory; the Hausdorff paradox and its descendants, Banach–Tarski and amenability.
• Analysis and algebra: the Hausdorff moment problem and summation methods, the Hausdorff–Young inequality, the Baker–Campbell–Hausdorff formula.
• Memory: the Hausdorff Center for Mathematics in Bonn bears his name, and the annotated Gesammelte Werke, edited by Egbert Brieskorn, Friedrich Hirzebruch, Walter Purkert, Reinhold Remmert and Erhard Scholz, includes his literary and philosophical writings alongside the mathematics.

Connections

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

Teachers and advisors

• Julius Robert Mayer 1891
• Heinrich Bruns 1891

Students

• Franz Hallenbach 1933
• Gustav Steinbach 1930
• Karl Bögel 1924

Influenced by

• Georg Cantor
• Maurice Fréchet
• Thomas Joannes Stieltjes
• Thomas Young
• Émile Borel
• Constantin Carathéodory
• Friedrich Nietzsche

Influenced

• Kazimierz Kuratowski
• Pavel Alexandrov
• Stefan Banach
• Wacław Sierpiński
• Abram Besicovitch

Colleagues and collaborators

• Issai Schur
• Richard Courant
• Eduard Study 1910-1913
• Otto Toeplitz

Friends

• Erich Bessel-Hagen

Also mentioned in this life

• Friedrich Wilhelm Bessel
• Joseph Fourier
• Mikhail Gromov
• Werner Heisenberg
• Henri Lebesgue
• Sophus Lie
• Frigyes Riesz
• Alfred Tarski
• John von Neumann
• Karl Weierstrass

Fields

• set theory 12 lives
• general topology 3 lives
• measure theory 5 lives
• analysis 20 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Felix Hausdorff, MacTutor History of Mathematics.
2. Felix Hausdorff, Wikipedia.
3. Felix Hausdorff, Mathematics Genealogy Project.
4. The namesake: Felix Hausdorff, University of Greifswald.
5. E. Brieskorn and W. Purkert, Felix Hausdorff: Biographie (Gesammelte Werke, Band IB), Springer, 2018.
6. E. Brieskorn (ed.), Felix Hausdorff zum Gedächtnis, Vieweg, 1996.

Mathematics

1. F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Ber. Verh. Sächs. Ges. Wiss. Leipzig 58 (1906).
2. F. Hausdorff, Grundzüge einer Theorie der geordneten Mengen, Math. Ann. 65 (1908).
3. F. Hausdorff, Grundzüge der Mengenlehre, Veit, Leipzig, 1914.
4. F. Hausdorff, Die Mächtigkeit der Borelschen Mengen, Math. Ann. 77 (1916).
5. F. Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1919).
6. F. Hausdorff, Summationsmethoden und Momentfolgen I, II, Math. Z. 9 (1921).
7. F. Hausdorff, Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen, Math. Z. 16 (1923).
8. F. Hausdorff, Gesammelte Werke, ed. E. Brieskorn, F. Hirzebruch, W. Purkert, R. Remmert, E. Scholz, Springer, 2001–.
9. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 3rd ed., 2014.
10. G. Tomkowicz and S. Wagon, The Banach–Tarski Paradox, Cambridge University Press, 2nd ed., 2016.
11. J. M. Plotkin (ed.), Hausdorff on Ordered Sets, American Mathematical Society, 2005.