Hausdorff studied at Leipzig University under Heinrich Bruns and Adolph Mayer, graduating in 1891 with a doctorate in applications of mathematics to astronomy. His thesis was titled Zur Theorie der astronomischen Strahlenbrechung Ⓣ and studied refraction and extinction of light in the atmosphere. He published four papers on astronomy and optics over the next few years and he submitted his habilitation thesis to Leipzig in 1895, also based on his research into astronomy and optics. His methods were based on those of Bruns who had developed his own method of determining refraction and extinction, based on an idea of Bessel.
However Hausdorff's main interests were in literature and philosophy and his circle of friends consisted almost entirely of writers and artists, such as the composer Max Reger, rather than scientists. He also seemed keen to make a name for himself in the world of literature, more so than in the world of mathematics, and he published his literary work under the pseudonym of Paul Mongré. In 1897 he published his first literary work Sant' Ilario: Thoughts from Zarathustra's Country which was a work of 378 pages. He published a philosophy book Das Chaos in kosmischer Auslese Ⓣ (1898) which is a critique of metaphysics contrasting the empirical with the transcendental world that he rejected. His next major literary work was a book of poem Ekstases Ⓣ (1900) which deals with nature, life, death and erotic passion, and in addition he wrote many articles on philosophy and literature. As Segal writes in :-
As the child of a wealthy family, he did not have to worry about making a career as a mathematician; for him, mathematics, both as research and as a subject to teach, was more an avocation than anything else.Hausdorff married Charlotte Sara Goldschmidt in Leipzig in 1899. Charlotte and her sister Edith were from Jewish parents but had converted to Lutheranism. Although still a Privatdozent, Hausdorff was well off, so marriage at this stage in his career presented no financial difficulties. In 1902 he was promoted to an extraordinary professorship of mathematics at Leipzig and turned down the offer of a similar appointment at Göttingen. This clearly indicates that at this time Hausdorff was keener to remain in his literary and artistic circle in Leipzig than he was to progress his career in mathematics. He continued his literary interests and in 1904 published a farce Der Arzt seiner Ehre Ⓣ. In many ways this marked the end of his literary interests but this farce was performed in 1912 and was very successful.
After 1904 Hausdorff began working in the area for which he is famous, namely topology and set theory. He introduced the concept of a partially ordered set and from 1901 to 1909 he proved a series of results on ordered sets. In 1907 he introduced special types of ordinals in an attempt to prove Cantor's continuum hypothesis. He also posed a generalisation of the continuum hypothesis by asking if 2 to the power ℵa was equal to ℵa+1. Hausdorff proved further results on the cardinality of Borel sets in 1916.
Hausdorff taught at Leipzig until 1910 when he went to Bonn. It was Study who in many ways motivated Hausdorff to become more involved in both mathematical research and also in developing his career in mathematics. Partly the lack of mathematical drive in his early career had been due to his extreme modesty, so his friendship with Study was an important factor in turning him towards important problems and his subsequent rise to fame. Having encouraged Hausdorff to move to Bonn, Study encouraged him to move again in 1913, this time to become an ordinary professorship at Greifswalf. A year later, in 1914, Hausdorff published his famous text Grundzüge der Mengenlehre Ⓣ which builds on work by Fréchet and others to created a theory of topological and metric spaces. Earlier results on topology fitted naturally into the framework set up by Hausdorff as Katetov explains in :-
[Hausdorff's] 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.The Grundzüge Ⓣ was republished in revised form in 1927 and 1937. The 1914 edition was reprinted in 1949 and 1965 by Chelsea, the 1927 edition was published in 1937 in Russian, and the 1937 edition was translated into English and also published by Chelsea in 1957.
Hausdorff returned to Bonn in 1921, by this time an emminent mathematician, and he worked there until 1935 when he was forced to retire by the Nazi regime. Although as early as 1932 he sensed the oncoming calamity of Nazism he made no attempt to emigrate while it was still possible. He swore the necessary oath to Hitler in November 1934 but by the following January a new law forced him to give up his position. He continued to undertake research in topology and set theory but the results could not be published in Germany. Certainly he wanted to continue research and wished to emigrate for in 1939 he wrote to Courant asking if he could find a research fellowship for him. Sadly Courant could not do so.
As a Jew his position became more and more difficult. In 1941 he was scheduled to go to an internment camp but managed to avoid being sent. Erich Bessel-Hagen, the only colleague from Bonn who kept in touch with Hausdorff after his forced retirement, wrote in a letter to a friend in the summer of 1941 (see  and ):-
I often had great anxiety about the Hausdorffs. Mrs Hausdorff was for a long time seriously ill from an old ailment - I don't know what it is. Scarcely was she over the worst than there came the agitation about the intended internment of the Jews. Here the procedure was mad. In the early part of the year, old nuns were forcibly driven out of a cloister on the Kreuzberg; these poor old women who never harmed anyone and only carried on a retiring life devoted to their pious usages ... Now all Jews still living in Bonn will be compulsorily interned in this stolen building; they must either auction their things, or place them for preservation in "faithful" hands.Bonn University requested that the Hausdorffs be allowed to remain in their home and this was granted. By October 1941 they were forced to wear the "yellow star" and around the end of the year they were informed that they would be sent to Cologne. Bessel-Hagen wrote that he knew this was (see  and ):-
... a preliminary to deportation to Poland. And what one hears concerning the accommodation and treatment of Jews there is completely unimaginable.They were not sent to Cologne but in January 1942 they were informed that they were to be interned in Endenich. Together with his wife and his wife's sister, he committed suicide on 26 January. He wrote to a friend on Sunday 25 January (see  and ):-
Dear Friend WollsteinOn the night of Sunday 25 January all three took barbiturates. Both Hausdorff and his wife Charlotte were dead by the morning of the 26 January. Edith, Charlotte's sister, survived for a few days in a coma before dying.
By the time you receive these lines, we three will have solved the problem in another way - in the way which you have continually attempted to dissuade us. ...
What has been done against the Jews in recent months arouses well-founded anxiety that we will no longer be allowed to experience a bearable situation. ...
Forgive us, that we still cause you trouble beyond death; I am convinced that you will do what you are able to do (and which perhaps is not very much). Forgive us also our desertion! We wish you and all our friends will experience better times
We have mentioned above Hausdorff's early work on astronomy, his work on philosophy, and his literature. We also mentioned his work on ordered sets and his masterpiece on set theory and topology Grundzüge der Mengenlehre Ⓣ (1914). Let us add that one famous paradoxical result, namely that half a sphere and one third of a sphere can be congruent to each other, is contained in this work (see  for details). Let us now examine other important contributions made by Hausdorff. In 1919 he introduced the notion of Hausdorff dimension in the seminal paper Dimension und äusseres Mass Ⓣ. The idea was a generalisation of one which had been introduced five years earlier by Carathéodory but Hausdorff realised that Carathéodory's construction made sense, and was useful, for defining fractional dimensions. Hausdorff's paper includes a proof that the dimension of the middle-third Cantor set is log 2/log 3. Chatterji writes :-
Within the mathematical work of Hausdorff the two publications devoted explicitly to measure theory occupy a significant place: they are not only important for measure theory but have also contributed fundamentally to its development. It is not well known that throughout his life Hausdorff had been interested in various fundamental problems of measure and integration theory and had made important contributions at different times. This becomes quite evident if one studies his lecture notes and other Nachlass papers.One such lecture course was given on probability theory by Hausdorff in Bonn in the summer of 1923. He studied the Gaussian law of errors, limit theorems and problems of moments, and set theory and the strong law of large numbers, which he based on measure theory.
Article by: J J O'Connor and E F Robertson
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