Hausdorff, Felix

(b. Breslau, Germany [now Wroc?aw, 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 ?s-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. Kat?tov