Felix Hausdorff

Work and reception

Hausdorff as philosopher and writer (Paul Mongré)

Hausdorff's volume of aphorisms, published in 1897, was his first work published under the pseudonym Paul Mongré. It is entitled Sant' Ilario: Thoughts from the landscape of Zarathustra. The subtitle plays first on the fact that Hausdorff had completed his book during a recovery stay on the Ligurian coast by Genoa and that in this same area, Friedrich Nietzsche wrote the first two parts of Thus Spoke Zarathustra; he also alludes to his spiritual closeness to Nietzsche. In an article on Sant' Ilario in the weekly paper Die Zukunft, Hausdorff acknowledged in expressis verbis his debt to Nietzsche.

Hausdorff was not trying to copy or even exceed Nietzsche. "Of Nietzsche imitation no trace", says a contemporary review. He follows Nietzsche in an attempt to liberate individual thinking, to take the liberty of questioning outdated standards. Hausdorff maintained critical distance to the late works of Nietzsche. In his essay on the book The Will to Power compiled from notes left in the Nietzsche Archive he says:

In Nietzsche glows a fanatic. His morality of breeding, erected on our present biological and physiological foundations of knowledge: that could be a world historical scandal against which the Inquisition and witch trials fade into harmless aberrations.

His critical standard he took from Nietzsche himself,

From the kind, modest, understanding Nietzsche and from the free spirit of the cool, dogma-free, unsystematic skeptic Nietzsche ...

In 1898—also under the pseudonym Paul Mongré—Hausdorff published an epistemological experiment titled Chaos in cosmic selection. The critique of metaphysics put forward in this book had its starting point in Hausdorff's confrontation with Nietzsche's idea of eternal recurrence. Ultimately, it is about destroying any kind of metaphysics. Of the world itself, of the transcendent core of the world—as Hausdorff puts it—we know nothing and we can know nothing. We must assume "the world itself" as undetermined and undeterminable, as mere chaos. The world of our experience, our cosmos, is the result of the selections that we have made and will always instinctively make according to our capacity for understanding. Seen from that chaos, all other frameworks, other cosmos, are conceivable. That is to say, from the world of our cosmos, one cannot draw any conclusions about the transcendent world.

In 1904, in the magazine The New Rundschau, Hausdorff's play appeared, the one-act play The doctor of his honor. It is a crude satire on the duel and on the traditional concepts of honor and nobility of the Prussian officer corps, which in the developing bourgeois society were increasingly anachronistic. The doctor of his honor was Hausdorff's most popular literary work. In 1914–1918 there were numerous performances in more than thirty cities. Hausdorff later wrote an epilogue to the play, but it was not performed at that time. Only in 2006 did this epilogue have its premier at the annual meeting of the German Mathematical Society in Bonn. In 2021 the first English translation of The doctor of his honor was published by The Hausdorff Center for Mathematics in Bonn.

Besides the works mentioned above, Hausdorff also wrote numerous essays that appeared in some of the leading literary magazines of the time. He also wrote a book of poems, Ecstasy (1900). Some of his poems were set to music by Austrian composer Joseph Marx.

Theory of ordered sets

Hausdorff's entrance into a thorough study of ordered sets was prompted in part by Cantor's continuum problem: where should the cardinal number ${\displaystyle \mathrm{\aleph }=2^{{\mathrm{\aleph }}_0}}$ be placed in the sequence ${\displaystyle \{{\mathrm{\aleph }}_{\alpha }\}}$? In a letter to Hilbert on 29 September 1904, he speaks of this problem, "it has plagued me almost like monomania". Hausdorff saw a new strategy to attack the problem in the set ${\displaystyle \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(T({\mathrm{\aleph }}_0))=\mathrm{\aleph }}$. Cantor had suspected ${\displaystyle \mathrm{\aleph }={\mathrm{\aleph }}_1}$, but had only been able to show that ${\displaystyle \mathrm{\aleph }\ge {\mathrm{\aleph }}_1}$. While ${\displaystyle {\mathrm{\aleph }}_1}$ is the "number" of possible well-orderings of a countable set, ${\displaystyle \mathrm{\aleph }}$ had now emerged as the "number" of all possible orders of such an amount. It was natural, therefore, to study systems that are more specific than orders, but more general than well-orderings. Hausdorff did just that in his first volume of 1901, with the publication of theoretical studies of "graded sets". However, we know from the results of Kurt Gödel and Paul Cohen that this strategy to solve the continuum problem is just as ineffectual as Cantor's strategy, which was aimed at generalizing the Cantor–Bendixson principle from closed sets to general uncountable sets.

In 1904 Hausdorff published the recursion named after him, which states that for each non-limit ordinal ${\displaystyle \mu }$ we have ${\displaystyle {\mathrm{\aleph }}_{\mu }^{{\mathrm{\aleph }}_{\alpha }}={\mathrm{\aleph }}_{\mu }{\mathrm{\aleph }}_{\mu -1}^{{\mathrm{\aleph }}_{\alpha }}.}$

This formula was, together with a later notion called cofinality introduced by Hausdorff, the basis for all further results for Aleph exponentiation. Hausdorff's excellent knowledge of recurrence formulas of this kind also empowered him to uncover an error in Julius König's lecture at the International Congress of Mathematicians in 1904 in Heidelberg. There König had argued that the continuum cannot be well-ordered, so its cardinality is not an Aleph at all, and thus caused a great stir. The fact that it was Hausdorff who clarified the mistake carries a special significance, since a false impression of the events in Heidelberg lasted for over 50 years.

In the years 1906–1909 Hausdorff did his groundbreaking and fundamental work on ordered sets. Of fundamental importance to the whole theory is the concept of cofinality, which Hausdorff introduced. If ${\displaystyle \alpha }$ is an ordinal, then a collection ${\displaystyle \mathcal{A}}$ of ordinals each less than ${\displaystyle \alpha }$ is cofinal in ${\displaystyle \alpha }$ if, for every ${\displaystyle \beta <\alpha }$, there is a ${\displaystyle \gamma \in \mathcal{A}}$ such that ${\displaystyle \beta \le \gamma }$. So, in a sense, a cofinal collection in ${\displaystyle \alpha }$ encompasses all of ${\displaystyle \alpha }$. Since a collection of objects from a well-ordered set is well-ordered under the original ordering, we can naturally assign a well-order to any such ${\displaystyle \mathcal{A}}$. As it turns out, any collection ${\displaystyle \mathcal{A}}$ of ordinals less than an ordinal ${\displaystyle \alpha }$ has an order type no greater than ${\displaystyle \alpha }$. An ordinal ${\displaystyle \alpha }$ is then called regular if any collection of ordinals cofinal in ${\displaystyle \alpha }$ has, in fact, the same order type as ${\displaystyle \alpha }$. In other words, ${\displaystyle \alpha }$ is regular if every collection of smaller ordinals that encompasses ${\displaystyle \alpha }$ has the same size, order-theoretically, as ${\displaystyle \alpha }$. Hausdorff's question, whether there are regular numbers which index a limit ordinal, was the starting point for the theory of inaccessible cardinals. Hausdorff had already noticed that such numbers, if they exist, must be of "exorbitant size".

The following theorem due to Hausdorff is also of fundamental importance: for each unbounded and ordered dense set ${\displaystyle A}$ there are two uniquely determined regular initial numbers ${\displaystyle {\omega }_{\xi },{\omega }_{\eta }}$ so that ${\displaystyle A}$ is cofinal with ${\displaystyle {\omega }_{\xi }}$ and coinitial with ${\displaystyle {\omega }_{\eta }^{\ast }}$ (where * denotes the inverse order). This theorem provides, for example, a technique to characterize elements and gaps in ordered sets.

If ${\displaystyle W}$ is a predetermined set of characters (element and gap characters), the question arises whether there are ordered sets whose character set is exactly ${\displaystyle W}$. One can easily find a necessary condition for ${\displaystyle W}$, but Hausdorff was also able to show that this condition is sufficient. For this one needs a rich reservoir of ordered sets, which Hausdorff was also able to create with his theory of general products and powers. In this reservoir can be found interesting structures like the Hausdorff ${\displaystyle {\eta }_{\alpha }}$ normal-types, in connection with which Hausdorff first formulated the generalized continuum hypothesis. Hausdorff's ${\displaystyle {\eta }_{\alpha }}$-sets formed the starting point for the study of the important model theory of saturated structure.

Hausdorff's general products and powers of cardinalities led him to study the concept of partially ordered set. The question of whether any ordered subset of a partially ordered set is contained in a maximal ordered subset was answered in the positive by Hausdorff using the well-ordering theorem. This is the Hausdorff maximal principle, which follows from either the well-ordering theorem or the axiom of choice, and as it turned out, is also equivalent to the axiom of choice.

Writing in 1908, Arthur Moritz Schoenflies found in his report on set theory that the newer theory of ordered sets (i.e., that which occurred after Cantor's extensions) was almost exclusively due to Hausdorff.

The "Magnum Opus": "Principles of set theory"

According to previous notions, set theory included not only the general set theory and the theory of sets of points, but also dimension and measure theory. Hausdorff's textbook was the first to present all of set theory in this broad sense, systematically and with full proofs. Hausdorff was aware of how easily the human mind can err while also seeking for rigor and truth, so in the preface of his work he promises:

… to be as economical as possible with the human privilege of error.

This book went far beyond its masterful portrayal of already-known concepts. It also contained a series of important original contributions by the author.

The first few chapters deal with the basic concepts of general set theory. In the beginning Hausdorff provides a detailed set algebra with some pioneering new concepts (differences chain, set rings and set fields, ${\displaystyle \delta }$- and ${\displaystyle \sigma }$-systems). The introductory paragraphs on sets and their connections included, for example, the modern set-theoretic notion of functions. Chapters 3 to 5 discussed the classical theory of cardinal numbers, order types and ordinals, and in the sixth chapter "Relations between ordered and well-ordered sets" Hausdorff presents, among other things, the most important results of his own research on ordered sets.

In the chapters on "point sets"—the topological chapters—Hausdorff developed for the first time, based on the known neighborhood axioms, a systematic theory of topological spaces, where in addition he added the separation axiom later named after him. This theory emerges from a comprehensive synthesis of earlier approaches of other mathematicians and Hausdorff's own reflections on the problem of space. The concepts and theorems of classical point set theory ${\displaystyle {\mathbb{R}}^n}$ are—as far as possible—transferred to the general case, and thus become part of the newly created general or set-theoretic topology. But Hausdorff not only performed this "translation work", but he also developed basic construction methods of topology such as core formation (open core, self-dense core) and shell formation (closure), and he works through the fundamental importance of the concept of an open set (called "area" by him) and of the concept of compactness introduced by Fréchet. He also founded and developed the theory of the connected set, particularly through the introduction of the terms "component" and "quasi-component".

With the first Hausdorff countability axiom, and eventually the second, the considered spaces were gradually further specialized. A large class of spaces satisfying the countable first axiom are metric spaces. They were introduced in 1906 by Fréchet under the name "classes (E)". The term "metric space" comes from Hausdorff. In Principles, he developed the theory of metric spaces and systematically enriched it through a series of new concepts: Hausdorff metric, complete, total boundedness, ${\displaystyle \rho }$-connectivity, reducible sets. Fréchet's work is not particularly famous; only through Hausdorff's Principles did metric spaces become common knowledge to mathematicians.

The chapter on illustrations and the final chapter of Principles on measure and integration theory are enriched by the generality of the material and the originality of presentation. Hausdorff's mention of the importance of measure theory for probability had great historical effect, despite its laconic brevity. One finds in this chapter the first correct proof of the strong law of large numbers of Émile Borel. Finally, the appendix contains the single most spectacular result of the whole book, namely Hausdorff's theorem that one cannot define a volume for all bounded subsets of ${\displaystyle {\mathbb{R}}^n}$ for ${\displaystyle n\ge 3}$. The proof is based on Hausdorff's paradoxical ball decomposition, whose production requires the axiom of choice.

During the 20th century, it became the standard to build mathematical theories on axiomatic set theory. The creation of axiomatically founded generalized theories, such as general topology, served among other things to single out the common structural core for various specific cases or regions and then set up an abstract theory, which contained all these parts as special cases. This brought a great success in the form of simplification and harmonization, and ultimately brought with itself an economy of thought. Hausdorff himself highlighted this aspect in the Principles. In the topological chapter, the basic concepts are methodologically a pioneering effort, and they paved the way for the development of modern mathematics.

Principles of set theory appeared in April 1914, on the eve of the First World War, which dramatically affected scientific life in Europe. Under these circumstances, the effects Hausdorff's book on mathematical thought would not be seen for five to six years after its appearance. After the war, a new generation of young researchers set forth to expand on the abundant suggestions that were included in this work. Undoubtedly, topology was the primary focus of attention. The journal Fundamenta Mathematicae, founded in Poland in 1920, played a special role in the reception of Hausdorff's ideas. It was one of the first mathematical journals with special emphasis on set theory, topology, the theory of real functions, measure and integration theory, functional analysis, logic, and foundations of mathematics. Across this spectrum, a special focus was placed on topology. Hausdorff's Principles was cited in the very first volume of Fundamenta Mathematicae, and through citation counting its influence continued at a remarkable rate. Of the 558 works (Hausdorff's own three works not included), which appeared in the first twenty volumes of Fundamenta Mathematicae from 1920 to 1933, 88 of them cite Principles. One must also take into account the fact that, as Hausdorff's ideas became increasingly commonplace, so too were they used in a number of works that did not cite them explicitly.

The Russian topological school, founded by Paul Alexandroff and Paul Urysohn, was based heavily on Hausdorff's Principles. This is shown by the surviving correspondence in Hausdorff's Nachlass with Urysohn, and especially Alexandroff and Urysohn's Mémoire sur les multiplicités Cantoriennes, a work the size of a book, in which Urysohn developed dimension theory and Principles is cited no fewer than 60 times.

After the Second World War there was a strong demand for Hausdorff's book, and there were three reprints at Chelsea from 1949, 1965 and 1978.

Descriptive set theory, measure theory and analysis

In 1916, Alexandroff and Hausdorff independently solved the continuum problem for Borel sets: Every Borel set in a complete separable metric space is either countable or has the cardinality of the continuum. This result generalizes the Cantor–Bendixson theorem that such a statement holds for the closed sets of ${\displaystyle {\mathbb{R}}^n}$. For linear ${\displaystyle G_{\delta }}$ sets William Henry Young had proved the result in 1903, for ${\displaystyle G_{\delta \sigma \delta }}$ sets Hausdorff obtained a corresponding result in 1914 in Principles. The theorem of Alexandroff and Hausdorff was a strong impetus for further development of descriptive set theory.

Among the publications of Hausdorff in his time at Greifswald the work Dimension and outer measure from 1919 is particularly outstanding. In this work, the concepts were introduced which are now known as Hausdorff measure and the Hausdorff dimension. It has remained highly topical and in later years has been one of the most cited mathematical works from the decade of 1910 to 1920.

The concept of Hausdorff dimension is useful for the characterization and comparison of "highly rugged quantities". The concepts of Dimension and outer measure have experienced applications and further developments in many areas such as in the theory of dynamical systems, geometric measure theory, the theory of self-similar sets and fractals, the theory of stochastic processes, harmonic analysis, potential theory, and number theory.

Significant analytical work of Hausdorff occurred in his second time at Bonn. In Summation methods and moment sequences I in 1921, he developed a whole class of summation methods for divergent series, which today are called Hausdorff methods. In Hardy's classic Divergent Series, an entire chapter is devoted to the Hausdorff method. The classical methods of Hölder and Cesàro proved to be special cases of the Hausdorff method. Every Hausdorff method is given by a moment sequence; in this context Hausdorff gave an elegant solution of the moment problem for a finite interval, bypassing the theory of continued fractions. In his paper Moment problems for a finite interval of 1923 he treated more special moment problems, such as those with certain restrictions for generating density ${\displaystyle \phi (x)}$, for instance ${\displaystyle \phi (x)\in L^p[0,1]}$. Criteria for solvability and decidability of moment problems occupied Hausdorff for many years, as hundreds of pages of handwritten notes in his Nachlass attest.

A significant contribution to the emerging field of functional analysis in the 1920s was Hausdorff's extension of the Riesz-Fischer theorem to ${\displaystyle L^p}$ spaces in his 1923 work An extension of Parseval's theorem on Fourier series. He proved the inequalities now named after him and W.H. Young. The Hausdorff–Young inequalities became the starting point of major new developments.

Hausdorff's book Set Theory appeared in 1927. This was declared as a second Edition of Principles, but it was actually a completely new book. Since the scale was significantly reduced due to its appearance in Goschen's teaching library, large parts of the theory of ordered sets and measures and integration theory were removed. In its preface, Hausdorff writes, "Perhaps even more than these deletions the reader will regret the most that, to further save space in point set theory, I have abandoned the topological point of view through which the first edition has apparently acquired many friends, and focused on the simpler theory of metric spaces".

In fact, this was an explicit regret of some reviewers of the work. As a kind of compensation Hausdorff showed for the first time the then-current state of descriptive set theory. This fact assured the book almost as intense a reception as Principles, especially in Fundamenta Mathematicae. As a textbook it was very popular. In 1935 there was an expanded edition published, and this was reprinted by Dover in 1944. An English translation appeared in 1957 with reprints in 1962 and 1967.

There was also a Russian edition (1937), although it was only partially a faithful translation, and partly a reworking by Alexandroff and Kolmogorov. In this translation the topological point of view again moved to the forefront. In 1928 a review of Set Theory was written by Hans Hahn, who perhaps had the danger of German antisemitism in his mind as he closed his discussion with the following sentence:

An exemplary depiction in every respect of a difficult and thorny area, a work on par with those which have carried the fame of German science throughout the world, and such that all German mathematicians may be proud of.

His last works

In 1938, Hausdorff's last work Extension of a continuous map showed that a continuous function from a closed subset ${\displaystyle F}$ of a metric space ${\displaystyle E}$ can be extended to all of ${\displaystyle E}$ (although the image may need to be extended). As a special case, every homeomorphism from ${\displaystyle F}$ can be extended to a homeomorphism from ${\displaystyle E}$. This work continued research from earlier years. In 1919, in About semi-continuous functions and their generalization, Hausdorff had, among other things, given another proof of the Tietze extension theorem. In 1930, in Extending a homeomorphism, he showed the following: Let ${\displaystyle E}$ be a metric space, ${\displaystyle F\subseteq E}$ a closed subset. If ${\displaystyle F}$ is given a new metric without changing the topology, this metric can be extended to the entire space without changing the topology. The work Graded spaces appeared in 1935, where Hausdorff discussed spaces which fulfilled the Kuratowski closure axioms up to the axiom of idempotence. These spaces are often also called closure spaces, and Hausdorff used them to study relationships between the Fréchet limit spaces and topological spaces.

Hausdorff as name-giver

The name Hausdorff is found throughout mathematics. Among others, these concepts were named after him:

• Hausdorff completion
• Hausdorff convergence
• Hausdorff density
• Hausdorff dimension
• Hausdorff distance
• Hausdorff gap
• Hausdorff maximal principle
• Hausdorff measure
• Hausdorff metric
• Hausdorff moment problem
• Hausdorff paradox
• Hausdorff space
• Hausdorff–Young inequality
• Baker–Campbell–Hausdorff formula

In the universities of Bonn and Greifswald, these things were named in his honor:

• the Hausdorff Center for Mathematics in Bonn,
• the Hausdorff Research Institute for Mathematics in Bonn, and
• the Felix Hausdorff Internationale Begegnungszentrum in Greifswald.

Besides these, in Bonn there is the Hausdorffstraße (Hausdorff Street), where he first lived. (Haus-Nr. 61). In Greifswald there is a Felix-Hausdorff–Straße, where the Institutes for Biochemistry and Physics are located, among others. Since 2011, there is a "Hausdorffweg" (Hausdorff-Way) in the middle of Leipziger Ortsteil Gohlis.

The Asteroid 24947 Hausdorff was named after him.

Writings

As Paul Mongré

Only a selection of the essays that appeared in text are shown here.

• Sant'Ilario. Gedanken aus der Landschaft Zarathustras. Verlag C. G. Naumann, Leipzig 1897.
• Das Chaos in kosmischer Auslese — Ein erkenntniskritischer Versuch. Verlag C. G. Naumann, Leipzig 1898; Reprinted with foreword by Max Bense: Baden-Baden: Agis-Verlag 1976, ISBN 3-87007-013-7
• Massenglück und Einzelglück. Neue Deutsche Rundschau (Freie Bühne) 9 (1), (1898), S. 64–75.
• Das unreinliche Jahrhundert. Neue Deutsche Rundschau (Freie Bühne) 9 (5), (1898), S. 443–452.
• Ekstasen. Volume of poetry. Verlag H. Seemann Nachf., Leipzig 1900.
• Der Wille zur Macht. In: Neue Deutsche Rundschau (Freie Bühne) 13 (12) (1902), S. 1334–1338.
• Max Klingers Beethoven. Zeitschrift für bildende Kunst, Neue Folge 13 (1902), S. 183–189.
• Sprachkritik Neue Deutsche Rundschau (Freie Bühne) 14 (12), (1903), S. 1233–1258.
• Der Arzt seiner Ehre, Groteske. In: Die neue Rundschau (Freie Bühne) 15 (8), (1904), S. 989-1013. New edition as: Der Arzt seiner Ehre. Komödie in einem Akt mit einem Epilog. With 7 portraits and woodcuts by Hans Alexander Müller after drawings by Walter Tiemann, 10 Bl., 71 S. Fifth printing by Leipziger Bibliophilen-Abends, Leipzig 1910. New edition: S. Fischer, Berlin 1912, 88 S. The Doctor of his Honor (English Translation of Der Arzt seiner Ehre, Groteske) by Volker Vetter and William P. Hausdorff, Hausdorff Center for Mathematics, Bonn 2021.

As Felix Hausdorff

• Beiträge zur Wahrscheinlichkeitsrechnung. Proceedings of the Royal Saxon Society for the Sciences at Leipzig. Math.-phys. Classe 53 (1901), S. 152–178.
• Über eine gewisse Art geordneter Mengen. Proceedings of the Royal Saxon Society for the Sciences at Leipzig. Math.-phys. Classe 53 (1901), S. 460–475.
• Das Raumproblem (Inaugural lecture at the University of Leipzig on 4. July 1903). Ostwald's Annals of Natural Philosophy 3 (1903), S. 1–23.
• Der Potenzbegriff in der Mengenlehre. Annual report of the DMV 13 (1904), S. 569–571.
• Untersuchungen über Ordnungstypen I, II, III. Proceedings of the Royal Saxon Society for the Sciences at Leipzig. Math.-phys.\ Klasse 58 (1906), S. 106–169.
• Untersuchungen über Ordnungstypen IV, V. Proceedings of the Royal Saxon Society for the Sciences at Leipzig. Math.-phys. Klasse 59 (1907), S. 84–159.
• Über dichte Ordnungstypen^{[link removed]}. Annual report of the DMV 16 (1907), S. 541–546.
• Grundzüge einer Theorie der geordneten Mengen^{[link removed]}. Math. Annalen 65 (1908), S. 435–505.
• Die Graduierung nach dem Endverlauf. Proceedings of the Royal Saxon Society for the Sciences at Leipzig. Math.-phys. Klasse 31 (1909), S. 295–334.
• Grundzüge der Mengenlehre. Verlag Veit & Co, Leipzig. 476 S. mit 53 Figuren. Further printings: Chelsea Pub. Co. 1949, 1965, 1978.
• Die Mächtigkeit der Borelschen Mengen^{[link removed]}. Math. Annalen 77 (1916), S. 430–437.
• Dimension und äußeres Maß. Math. Annalen 79 (1919), S. 157–179.
• Über halbstetige Funktionen und deren Verallgemeinerung^{[link removed]}. Math. Zeitschrift 5 (1919), S. 292–309.
• Summationsmethoden und Momentfolgen I^{[link removed]}, II.^{[link removed]} Math. Zeitschrift 9 (1921), I: S. 74-109, II: S. 280–299.
• Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen. Math. Zeitschrift 16 (1923), S. 163–169.
• Momentprobleme für ein endliches Intervall. Math. Zeitschrift 16 (1923), S. 220–248.
• Mengenlehre, second reworked edition. Verlag Walter de Gruyter & Co., Berlin. 285 S. with 12 figures.
• Erweiterung einer Homöomorphie (PDF; 389 kB) Fundamenta Mathematicae 16 (1930), S. 353–360.
• Mengenlehre, third edition. With an additional chapter and several appendices. Verlag Walter de Gruyter & Co., Berlin. 307 S. mit 12 Figuren. Nachdruck: Dover Pub. New York, 1944. Englisch edition: Set theory. Translated from the German by J. R. Aumann et al. Chelsea Pub. Co., New York 1957, 1962, 1967.
• Gestufte Räume. (PDF; 1,2 MB) Fundamenta Mathematicae 25 (1935), S. 486–502.
• Erweiterung einer stetigen Abbildung (PDF; 450 kB) Fundamenta Mathematicae 30 (1938), S. 40–47.
• Nachgelassene Schriften. 2 volumes. Ed.: G. Bergmann, Teubner, Stuttgart 1969. From the Nachlass, Volume I includes fascicles 510–543, 545–559, 561–577, Volume II fascicles 578–584, 598–658 (all fascicles given in facsimile).

Hausdorff on Ordered Sets. Trans. and Ed.: Jacob M. Plotkin, American Mathematical Society 2005.

Collected works

The "Hausdorff-Edition", edited by E. Brieskorn (†), F. Hirzebruch (†), W. Purkert (all Bonn), R. Remmert (†) (Münster) and E. Scholz (Wuppertal) with the collaboration of over twenty mathematicians, historians, philosophers and scholars, is an ongoing project of the North Rhine-Westphalian Academy of Sciences, Humanities and the Arts to present the works of Hausdorff, with commentary and much additional material. The volumes have been published by Springer-Verlag, Heidelberg. Nine volumes have been published with volume I being split up into volume IA and volume IB. See the website of the Hausdorff Project website of the Hausdorff Edition (German) for further information. The volumes are:

• Band IA: Allgemeine Mengenlehre. 2013, ISBN 978-3-642-25598-4.
• Band IB: Felix Hausdorff – Paul Mongré (Biographie). 2018, ISBN 978-3-662-56380-9.
• Band II: Grundzüge der Mengenlehre (1914). 2002, ISBN 978-3-540-42224-2
• Band III: Mengenlehre (1927, 1935); Deskriptive Mengenlehre und Topologie. 2008, ISBN 978-3-540-76806-7
• Band IV: Analysis, Algebra und Zahlentheorie. 2001, ISBN 978-3-540-41760-6
• Band V: Astronomie, Optik und Wahrscheinlichkeitstheorie. 2006, ISBN 978-3-540-30624-5
• Band VI: Geometrie, Raum und Zeit. 2020. ISBN 978-3-540-77838-7
• Band VII: Philosophisches Werk. 2004, ISBN 978-3-540-20836-5
• Band VIII: Literarisches Werk. 2010, ISBN 978-3-540-77758-8
• Band IX: Korrespondenz. 2012, ISBN 978-3-642-01116-0.