VI.68 Felix Hausdorff
b. Breslau, Germany (now Wroclaw, Poland), 1868;
d. Bonn, Germany, 1942
Set theory; topology
Hausdorff studied mathematics at Leipzig, Freiburg, and Berlin between 1887 and 1891, and then started research in applied mathematics at Leipzig under Heinrich Bruns. After his habilitation (1895) he taught first at Leipzig and then later at Bonn (1910-13, 1921-35) and Greifswald (1913-21). He is best known for his work in set theory and general topology, his magnum opus being Grundzüge der Mengenlehre (“Basic features of set theory”). It was published in 1914 and had second and third editions in 1927 and 1935. The second edition was so heavily revised in content, however, that it should really be considered a new book.
In his early work, Hausdorff concentrated on applied mathematics, mainly related to astronomy, in particular the refraction and extinction of light in the atmosphere. He had broad intellectual interests and moved in Nietzschean circles of artists and poets at Leipzig. Under the pseudonym Paul Mongré he wrote two long philosophical essays of which the more prominent was “Das Chaos in kosmischer Auslese” (“The chaos in cosmic selection”). Until 1904 he regularly contributed cultural critical essays to a renowned German intellectual review of the time, continuing to contribute, although less frequently, until 1912. He also published poems and a satirical play.
Hausdorff took up set theory at the turn of the century and gave his first lecture course on the topic in the summer semester of 1901 at Leipzig university. After his turn toward “Cantorianism” (set theory) he began deep and innovative research on order structures and their classification. Among the results of his early work in set theory are the Hausdorff recursion formula for exponentiation of cardinals and several contributions to the study of order structures (cofinality, etc.). Although Hausdorff did not pursue active research in the axiomatic foundation of set theory, he contributed important insights on transfinite numbers, in particular a characterization of what are now known as weakly inaccessible cardinals and his maximal chain principle, a form of ZORN’S LEMMA [III.1] that predated the latter and differed from it in formulation and intention.
His own contribution to the axiomatic method was oriented toward generalizing classical areas of mathematics and founding them on axiomatic principles within the framework of set theory. Hausdorff’s move to use set theory inside mathematics was seminal for the turn toward modern mathematics in the sense of the twentieth century, most prominently characterized by the BOURBAKI [VI.96] group. Best known in this respect are his axiomatization of general topology in terms of axioms for neighborhood systems, first published in the Grundzüge (1914), and the study of the properties of general, or more specialized, TOPOLOGICAL SPACES [III.90]. Less well-known (it remained unpublished until recently) was Hausdorff’s axiomatization of probability theory, which was presented in a lecture course of 1923 and which preceded KOLMOGOROV’S [VI.88] work in this area by about a decade. He also made important contributions to analysis and algebra. In algebra, he contributed to LIE THEORY [III.48] (via what is now called the Baker-Campbell-Hausdorff formula), while in analysis he developed summation methods for divergent series and also a generalization of the Riesz-Fischer theory.
Hausdorff’s central goals in using set theory were for applications to analytical disciplines such as function theory. Among his most important contributions in this respect, and of wide-ranging importance, was the concept of HAUSDORFF DIMENSION [III.17], which he introduced to give a notion of dimension to rather general sets (such as, for example, fractal-type sets).
Hausdorff realized that analytical questions of set theory were deeply connected to foundational questions. In 1916 he (and, independently, P. Alexandroff) showed that any uncountable BOREL SET [III.55] in the reals actually has the cardinality of the continuum. This was an important development of a strategy proposed by Cantor to clarify the continuum. Although this strategy did not finally contribute to the decisive results by Gödel and Cohen on the CONTINUUM HYPOTHESIS [IV.22 §5], it led to the development of an extended field of investigation in the border region between set theory and analysis, now dealt with in DESCRIPTIVE SET THEORY [IV.22 §9]. Hausdorff’s second edition of the Mengenlehre (1927) was the first monograph in this field.
After the rise to power of the Nazi regime, working conditions and life in general deteriorated more and more drastically for Hausdorff and others of Jewish origin. When Hausdorff, his wife Charlotte, and a sister of hers were ordered to leave their house for local internment in January 1942, they opted for suicide rather than suffering further persecution.
Further Reading
Brieskorn, E. 1996. Felix Hausdorff zum Gedächtnis. Aspekte seines Werkes. Braunschweig: Vieweg.
Hausdorff, F. 2001. Gesammelte Werke einschließlich der unter dem Pseudonym Paul Mongré erschienenen philosophischen und literarischen Schriften, edited by E. Brieskorn, F. Hirzebruch, W. Purkert, R. Remmert, and E. Scholz. Berlin: Springer.
Hausdorff’s voluminous unpublished work (his “Nachlass”) can be found online at www.aic.uni-wuppertal.de/fb7/hausdorff/findbuch.asp.