# Hausdorff, Felix

# HAUSDORFF, FELIX

(*b*. Breslau, Germany, 8 November 1868; *d*. Bonn, Germany, 26 January 1942),

*mathematics, philosophy, literature*.

Hausdorff studied at Leipzig, Freiburg, and Berlin between 1887 and 1891 and started research in applied mathematics. After his *habilitation* (1895) he taught at the University of Leipzig and a local commercial school. He moved in a milieu of Leipzig intellectuals and artists, strongly influenced by the early work of Friedrich Nietzsche (1844–1900), striving for a cultural modernization of late-nineteenth-century Germany.

Between 1897 and 1910 Hausdorff published two philosophical books, a poem collection, and a satirical theater play under the pseudonym Paul Mongré. He regularly contributed cultural critical essays to the *Neue Deutsche Rundschau*, a leading German intellectual journal of the time. In his second book, *Das Chaos in kosmischer Auslese* (1898; Chaos in cosmic selection), he radicalized Kantian transcendental idealism by a Nietzschean perspective and attempted to dissolve the belief in any determined a priori form of transcendent reality. In particular he reviewed the contemporary discussion on space and time, employing transformations of increasing generality. In the end he was led to considering transfinite Cantorian sets and their general set transformations as a mathematical expression for a completely unstructured reality, and thus to ‘‘transcendent nihilism.”

During this period, Hausdorff reoriented his mathematical work toward the new field of transfinite set theory. He gave one of the first lecture courses on this topic in summer 1901 and contributed important results to it, among others the *Hausdorff recursion* for aleph exponentiation (1904) and deep methods for the classification of order structures (confinality, gap types, general ordered products, and eta-alpha sets; 1906–1907). He employed a “naive” concept of set, but even so achieved an exceptionally high precision of argumentation. He contributed crucial insights into foundational questions, most importantly his maximal chain principle (related to but different from Zorn’s lemma), a characterization of weakly inaccessible cardinals (in present terminology), and the universality property for order structures of what he called “eta-alpha sets.” The latter became one of the roots of *saturated structures* in model theory of the 1960s. Moreover, Hausdorff hit on the importance of the *generalized continuum hypothesis* in these studies.

In summer 1910 Hausdorff started teaching at the University of Bonn and broadened his perspective on set theory as a general basis for mathematics. In early 1912 he found an axiomatic characterization of topological spaces by neighborhood systems and started to compose a monograph on “basic features of set theory” (*Grundzüge der Mengenlehre*). It was finished two years later (1914b), after he was called in 1913 to a full professorship at the University of Greifswald.

In this book, Hausdorff showed how set theory could be used as a working frame for mathematics more broadly. While he introduced set theory in a nonaxiomatic style, although with extraordinary precision, topological spaces and measure theory were given an axiomatic presentation. In part two Hausdorff published his neighborhood axioms found two years earlier, introduced separation and countability axioms, and studied connectivity properties. This part of the book contained the first comprehensive treatment of the theory of metric spaces, initiated by Maurice-René Fréchet in 1906. It laid the basis for an important part of general topology of the twentieth century.

In part three he provided a lucid introduction to measure theory, building on the work of Émile Borel and Henri-Léon Lebesgue. In a paper published shortly before the book (also in an appendix to it) Hausdorff gave a negative answer to Lebesgue’s question, whether a (finitely) additive content function invariant under congruences can be defined on *all* subsets of Euclidean three-space (1914a). His peculiar use of the axiom of choice became the starting point for the later paradoxical constructions of measure theory by Stefen Banach and Alfred Tarski.

The *Grundzüge* became influential only after World War I, most strongly in the rising schools of modern mathematics in Poland, around the journal *Fundamenta Mathematicae*, and in the Soviet Union mainly among Nikolai N. Luzin’s students around Pavel Alexandrov. The *Grundzüge* became one of the founding documents of mathematical modernism in the 1920 and 1930s.

Already in the *Grundzüge* Hausdorff had started to study Borel sets. In 1916 he, and independently Alexandrov, could show that any infinite Borel set in a separable metrical space is countable or of cardinality of the continuum. That was an important step forward for a strategy proposed by Georg Cantor to clarify the continuum hypothesis. Although this goal could not be achieved along this road, it led to the development of an extended field of investigation on the border region between set theory and analysis, now dealt with in descriptive set theory.

On the other side, Hausdorff took up questions in real analysis, now informed by the new basic features of general set theory. His introduction of what came to be called Hausdorff measure and Hausdorff dimension (1919) became of long-lasting importance in the theory of dynamical systems, geometrical measure theory, and the study of “fractals.”

Other important technical contributions dealt with summation methods of infinite divergent series and a generalization of the Riesz-Fischer theorem, which established the well-known relation between function spaces and series of Fourier coefficients (1923). It opened the path for later developments in harmonic analysis on topogical groups. In a lecture course in 1923 Hausdorff introduced an axiomatic basis for probability theory, which anticipated Andrey Kolmogorov’s axiomatization of 1933.

When Hausdorff revised his magnum opus for a second edition, he rewrote the parts on descriptive set theory and topological spaces completely, extending the first part considerably and concentrating the second one on metrical spaces. As other books on general set and general topology had appeared in the meantime, he omitted these parts for the so-called second edition, which became essentially a new book (1927).

In 1921 Hausdorff had returned to Bonn, now as a full professor and colleague of Eduard Study and (some years later) Otto Toeplitz. While he was still regularly emeritated in early 1935, general life and working conditions deteriorated drastically for Hausdorff and other people of Jewish origin, after the rise to power of the Nazi regime. His attempt at emigration came too late. When Hausdorff, his wife Charlotte, and a sister of hers were ordered to leave their house for a local internment camp in January 1942, they opted for suicide rather than suffering further persecution. His scientific *Nachlass* was handed over to a local friend. It survived the end of the war with only minor damage.

## BIBLIOGRAPHY

*Hausdorff’s complete works are currently being published under the title* Gesammelte Werke: Einschlieβlich der unter dem Pseudonym Paul Mongré erschienenen philosophischen und literarischen Schriften und ausgewählte Texte aus dem Nachlass,*edited by Egbert Brieskorn, Friedrich Hirzebruch, Walter Purkert, et al. (Berlin: Springer: 2001–). Nine volumes are planned. Volumes, with year of publication, are as follows: vol. 2 (2002), vol. 3 (2007), vol. 4 (2001), vol. 5 (2005), and vol. 7 (2004). Reference is made below to works that have been published in the above volumes (* Werke II, Werke IV, *and so forth)*.

### WORKS BY HAUSDORFF

(Under pseudonym Paul Mongré.) *Das Chaos in kosmischer Auslese—Ein erkenntniskritischer Versuch*. Leipzig, Germany: Naumann, 1898. In *Werke VII*, pp. 587–807.

“Der Potenzbegriff in der Mengenlehre.” *Jahresberichte der Deutschen Mathematiker-Vereinigung* 13 (1904): 569–571. English translation in *Hausdorff on Ordered Sets*, edited by Jacob M. Plotkin, pp. 31–33. Providence, RI: American Mathematical Society, 2005.

“Untersuchungen über Ordnungstypen I–V.” *Berichte Verhandlungen Sächsische Gesellschaft der Wissenschaften Leipzig, Math-Phys. Klasse* 58 (1906): 106–169, and 59 (1907): 84–159. English translation in *Hausdorff on Ordered Sets*, edited by Jacob M. Plotkin, pp. 35–171. Providence, RI: American Mathematical Society, 2005.

“Bemerkung über den Inhalt von Punktmengen.” *Mathematische Annalen* 75 (1914a): 428–433. In *Werke IV*, pp. 3–10.

*Grundzüge der Mengenlehre*. Leipzig, Germany: Veit, 1914b. In *Werke II*, pp. 91–576.

“Dimension und äuβeres Maβ.” *Mathematische Annalen* 79 (1919): 157–179. In *Werke IV*, pp. 21–43.

“Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen.” *Mathematische Zeitschrift* 16 (1923): 163–169. In *Werke IV*, pp. 173–181.

*Vorlesung Wahrscheinlichkeitsrechnung*, Bonn 1923, *Nachlass* Fasz. 64. In *Werke V*, pp. 595–723.

*Mengenlehre*. Berlin: Gruyter, 1927. In *Werke III*.

*Nachgelassene Schriften*. 2 vols. Edited by Günter Bergmann. Stuttgart, Germany: Teubner, 1969.

“Nachlass.” Handschriftenabteilung Universitätsbibliothek Bonn. Findbuch (catalog). Available from http://www.aic.uni-wuppertal.de/fb7/hausdorff/findbuch.asp.

### OTHER SOURCES

Brieskorn, Egbert, ed. *Felix Hausdorff zum Gedächtnis*. Braunschweig, Germany: Vieweg, 1996.

Chatterji, Srishti D. “Measure and Integration Theory.” In *Werke II*, pp. 788–800.

Eichhorn, Eugen, and Ernst-Jochen Thiele, eds. *Vorlesungen zum Gedenken an Felix Hausdorff*. Berlin: Heldermann, 1994.

Epple, Moritz, Horst Herrlich, Mirek Huŝek, et al. “Zum Begriff des topologischen Raumes.” In *Werke II*, pp. 675–744. Also available from http://hausdorff-edition.de/media/pdf/Topologischer_Raum.pdf.

Felgner, Ulrich. “Die Hausdorffsche Theorie der eta-alpha-Mengen und ihre Wirkungsgeschichte.” In *Werke II*, pp. 645–674. Also available from http://hausdorff-edition.de/media/pdf/Eta_Alpha.pdf.

Koepke, Peter, and Vladimir Kanovei. “Deskriptive Mengenlehre in Hausdorffs Grundzügen der Mengenlehre.” In *Werke II*, pp. 773–787.

Moore, Gregory. *Zermelo’s Axiom of Choice: Its Origins, Development, and Influence*. New York: Springer, 1982.

Neuenschwander, Erwin. “Felix Hausdorffs letzte Lebensjahre nach Dokumenten aus dem Bessel-Hagen Nachlaβ.” In *Felix Hausdorff zum Gedächtnis*, edited by Egbert Brieskorn, 253–270. Braunschweig, Germany: Vieweg, 1996.

Plotkin, Jacob M. *Hausdorff on Ordered Sets*. Providence, RI: American Mathematical Society, 2005.

Purkert, Walter. “Grundzüge der Mengenlehre: Historische Einführung.” In *Werke II*, pp. 1–90. Also available from http://hausdorff-edition.de/media/pdf/HistEinfuehrung.pdf.

Stegmaier, Walter. “Ein Mathematiker in der Landschaft Zarathustras: Felix Hausdorf als Philosoph.” *Nietzsche Studien* 31 (2002): 195–240. Similarly editor’s introduction in *Werke VII*, pp. 1–83. Also available from http://hausdorff-edition.de/media/pdf/Einleitung.pdf.

Vollhardt, Friedrich, and Udo Roth. “Die Signifikanz des Auβenseiters: Der Mathematiker Felix Hausdorff und die Weltanschauungsliteratur um 1900.” In *Literatur und Wissen(schaften), 1890–1935*, edited by Christine Maillard and Michael Titzmann, 213–234. Stuttgart, Germany: Metzler, 2002.

*Erhard Scholz*

# Hausdorff, Felix

# Hausdorff, Felix

(*b*. Breslau, Germany [now Wrocław, Poland], 8 November 1868; *d*. Bonn, Germany, 26 January 1942)

*mathematics*.

Hausdorff’s father was a wealthy merchant. After finishing his secondary education in Leipzig, Hausdorff studied mathematics and astronomy at Leipzig, Freiburg, and Berlin. He graduated from Leipzig in 1891 and five years later became a *Dozent* there. Until 1902, when he was appointed professor at Leipzig, he lived independently and devoted himself to a wide range of interests. From 1891 to 1896 he published four papers in astronomy and optics, and in the following years several papers in various branches of mathematics. His main interests, though, were philosophy and literature, and his friends were mainly artists and writers. Under the pen name Dr. Paul Mongre he published two books of poems and aphorisms; a philosophical book, *Das Chaos in kosmischer Auslese* (1898); and a number of philosophical essays and articles on literature. In 1904 he published a farce, *Der Arzt seiner Ehre*, which was produced in 1912 and had considerable success.

In 1902 Hausdorff became associate professor at Leipzig. From that time, mainly after 1904, he seems to have dealt more with set theory, at the same time gradually decreasing his nonscientific writing. In 1910 he went to Bonn as associate professor and there wrote the monograph *Grundzüge der Mengenlehre*, which appeared in 1914. In 1913 Hausdorff became full professor at Greifswald and in 1921 returned to Bonn, where he was active until his forced retirement in 1935. Even then he continued working on set theory and topology, although his work was published only outside Germany. As a Jew he was scheduled to be sent to an internment camp in 1941. It was temporarily avoided; but when internment became imminent, Hausdorff committed suicide with his wife and her sister on 26 January 1942.

Hausdorff’s scientific activity contributed greatly to several fields of mathematics. In mathematical analysis he proved important theorems concerning summation methods, properties of moments, and Fourier coefficients (1921). In algebra he derived and investigated the so-called symbolic exponential formula (1906). He introduced and investigated a very important class of measures and, in connection with them, a kind of dimension which may assume arbitrary nonnegative values (1919). Both are now named for Hausdorff and are applied in particular to examination of fine properties of numerical sets.

Hausdorff’s main work was in topology and set theory. Various definitions of topological spaces and related concepts had been given, mainly by Maurice Fréchet, before Hausdorff’s *Grundzüge der Mengenlehre appeared*. The interrelations of these different approaches had not been completely recognized; and no clear way had been known to effect a gradual transition from very general spaces to those similar to spaces actually occurring in analysis and geometry.

In the *Grundzüge*, Hausdorff took a decisive step in this direction. His broad approach, his aesthetic feeling, and his sense of balance may have played a substantial part. He succeeded in creating a theory of topological and metric spaces into which the previous results fitted well, and he enriched it with many new notions and theorems. From the modern point of view, the *Grundzüge* contained, in addition to other special topics, the beginnings of the theories of topological and metric spaces, which are now included in all textbooks on the subject. In the *Grundzüge*, these theories were laid down in such a way that a strong impetus was provided for their further development. Thus, Hausdorff can rightly be considered the founder of general topology and of the general theory of metric spaces.

The *Grundzüge* is a very rare case in mathematical literature: the foundations of a new discipline are laid without the support of any previously published comprehensive work.

Hausdorff’s work in topology and set theory has also brought about a number of separate results of primary importance: in topology, a detailed investigation into the basic properties of general closure spaces (1935); in general set theory, the so-called Hausdorff maximal principle (stated, although not explicitly, in the *Grundzüge*), the introduction of partially ordered sets, and several theorems on ordered sets (1906–1909); in descriptive set theory, the theorem on the cardinality of Borel sets (1916; proved independently by P. S. Alexandrov in the same year) and the introduction of the δs-operations, now often called Hausdorff operations (1927).

Hausdorff’s manuscripts have not yet been fully prepared for publication, but they are not likely to provide any new scientific results.

## BIBLIOGRAPHY

I. Original Works. Hausdorff’s major work is *Grundzüge der Mengenlehre* (Leipzig, 1914); the 2nd ed., entitled *Mengenlehre* (Leipzig, 1927), is in fact a new book. The Russian trans. (Moscow, 1935) is a revised combination of both. Hausdorff’s MSS are being published under the title *Nachgelassene Schriffen*, Günter Bergmann, ed.; vols. I and II appeared in Stuttgart in 1969. Numerous papers are in *Fundamenta mathematicae, Mathematische Annalen*, and *Mathematische Zeitschrift*.

II. Secondary Literature. A short biography, an analysis of Hausdorff’s work, a list of scientific papers, and a survey of his MSS are in M. Dierkesmann *et al*., “Felix Hausdorff zum Gedächtnis,” in *Jahresberichte der Deutschen Mathematikervereinigung*, **69** (1967), 51–76. An article on Hausdorff by W. Krull in “Bonner Gelehrte,” in *Beiträge zur Geschichte der Wissenschaften in Bonn*, Mathematik und Naturwissenschaften (1970), pp. 54–69, contains a short biography including an account of Hausdorff’s activity outside mathematics, and a detailed analysis of the *Grundzüge*. See also W. Krull, “Felix Hausdorff,” in *Neue deutsche Biographic*, VIII (Berlin, 1969), 111–112. The pref. to vol. I of *Nachgelassene Schriften* includes material on Hausdorff as a university teacher and a number of short excerpts from his correspondence.

M. KatĚtov

# Hausdorff, Felix

# HAUSDORFF, FELIX

**HAUSDORFF, FELIX** (1868–1942), German mathematician. Hausdorff was born in Breslau, and was professor of mathematics at Greifswald from 1913 to 1921 and at Bonn from 1921 until his retirement in 1935. Together with his wife he committed suicide in 1942 in order to avoid the deportation order of the Gestapo. Hausdorff was an authority on set theory and its applications to sets of points and real analysis. His textbook *Mengenlehre* (Leipzig, 1935) is recognized as one of the great classics of set theory. The depth and simplicity of his research into fundamental problems was a source of inspiration in the rapid development of modern mathematics. Hausdorff was devoted to music and literature and published belles lettres under the pen name of Paul Mongré.

## bibliography:

*Poggendorff's biographisch-literarisches Handwoerterbuch der exakten Naturwissenschaften*, 7a (1958), 402.

[Barry Spain]

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