Felix Hausdorff
Felix Hausdorff | |
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University of Leipzig | |
Known for | |
Spouse | Charlotte Hausdorff (1873-1942) |
Scientific career | |
Fields | University of Leipzig |
Thesis | Zur Theorie der astronomischen Strahlenbrechung (1891) |
Doctoral advisor |
|
Felix Hausdorff (
Life became difficult for Hausdorff and his family after the
Life
Childhood and youth
Hausdorff's father, the
Hausdorff's mother, Hedwig (1848–1902), who is also referred to in various documents as Johanna, came from the Jewish Tietz family. From another branch of this family came
From 1878 to 1887 Felix Hausdorff attended the Nicolai School in Leipzig, a facility that had a reputation as a hotbed of humanistic education. He was an excellent student, class leader for many years and often recited self-written Latin or German poems at school celebrations.
In his later years of high school, choosing a main subject of study was not easy for Hausdorff. Magda Dierkesmann, who was often a guest in the home of Hausdorff in the years 1926–1932, reported in 1967 that:
His versatile musical talent was so great that only the insistence of his father made him give up his plan to study music and become a composer.
He decided to study the natural sciences, and in his graduating class of 1887 he was the only one who achieved the highest possible grade.
Degree, doctorate and Habilitation
From 1887 to 1891 Hausdorff studied
In later semesters of his studies, Hausdorff was close to Heinrich Bruns (1848–1919). Bruns was professor of astronomy and director of the observatory at the University of Leipzig. Under his supervision, Hausdorff graduated in 1891 with a work on the theory of astronomical refraction of light in the atmosphere. Two publications on the same subject followed, and in 1895 his Habilitation also followed with a thesis on the absorbance of light in the atmosphere. These early astronomical works of Hausdorff, despite their excellent mathematical formulation, were ultimately of little importance to the scientific community. For one, the underlying idea of Bruns was later shown to not be viable (there was a need for refraction observations near the astronomical horizon, and as Julius Bauschinger would show, this could not be obtained with the required accuracy). And further, the progress in the direct measurement of atmospheric data (from
Lecturer in Leipzig
After his Habilitation, Hausdorff became a lecturer at the University of Leipzig where he began extensive teaching in a variety of mathematical areas. In addition to teaching and research in mathematics, he also pursued his literary and philosophical inclinations. A man of varied interests, he often associated with a number of famous writers, artists and publishers such as Hermann Conradi, Richard Dehmel, Otto Erich Hartleben, Gustav Kirstein, Max Klinger, Max Reger and Frank Wedekind. The years of 1897 to 1904 mark the high point of his literary and philosophical creativity, during which time 18 of his 22 pseudonymous works were published, including a book of poetry, a play, an epistemological book and a volume of aphorisms.
In 1899 Hausdorff married Charlotte Goldschmidt, the daughter of Jewish doctor Siegismund Goldschmidt. Her stepmother was the famous suffragist and preschool teacher Henriette Goldschmidt. Hausdorff's only child, his daughter Lenore (Nora), was born in 1900; she survived the era of National Socialism and enjoyed a long life, dying in Bonn in 1991.
First professorship
In December 1901 Hausdorff was appointed as adjunct associate professor at the University of Leipzig. An often-repeated factoid, that Hausdorff got a call from Göttingen and rejected it, cannot be verified and is most likely wrong. After considering Hausdorff's application to Leipzig, the Dean Kirchner felt compelled to make the following addition to the very positive vote from his colleagues, written by Heinrich Bruns:
The faculty, however, considers itself obliged to report to the Royal Ministry that the above application, considered on November 2nd of this year when a faculty meeting had taken place, was not accepted by all, but with 22 votes to 7. The minority was opposed, because Dr. Hausdorff is of the Mosaic faith.[4]
This quote emphasizes the undisguised
After his Habilitation, Hausdorff wrote other works on
For the summer semester of 1910 Hausdorff was appointed as professor to the University of Bonn. There he began a lecture series on set theory, which he substantially revised and expanded for the summer semester of 1912.
In the summer of 1912 he also began work on his magnum opus, the book Basics of set theory. It was completed in Greifswald, where Hausdorff had been appointed for the summer semester as full professor in 1913, and was released in April 1914.
The University of Greifswald was the smallest of the Prussian universities. The mathematical institute there was also small; during the summer of 1916 and the winter of 1916/17, Hausdorff was the only mathematician in Greifswald. This meant that he was almost fully occupied in teaching basic courses. It was thus a substantial improvement for his academic career when Hausdorff was appointed in 1921 to Bonn. There he was free to teach about wider ranges of topics, and often lectured on his latest research. He gave a particularly noteworthy lecture on probability theory (NL Hausdorff: Capsule 21: Fasz 64) in the summer semester of 1923, in which he grounded the theory of probability in measure-theoretic axiomatic theory, ten years before A. N. Kolmogorov's "Basic concepts of probability theory" (reprinted in full in the collected works, Volume V). In Bonn, Hausdorff was friends and colleagues with Eduard Study, and later with Otto Toeplitz, who were both outstanding mathematicians.
Under the Nazi dictatorship and suicide
After the takeover by the
On March 31, 1935, after some back and forth, Hausdorff was finally given emeritus status. No words of thanks were given for his 40 years of successful work in the German higher education system.
His academic legacy shows that Hausdorff was still working mathematically during these increasingly difficult times, and continued to follow current developments of interest. He wrote, in addition to the expanded edition of his work on set theory, seven works on topology and descriptive set theory. These were published in Polish magazines: one in Studia Mathematica, the others in Fundamenta Mathematicae. He was supported at this time by Erich Bessel-Hagen, a loyal friend to the Hausdorff family who obtained books and magazines from the academic library, which Hausdorff was no longer allowed to enter.
A great deal is known about the humiliations to which Hausdorff and his family especially were exposed to after Kristallnacht in 1938. There are many sources, including the letters of Bessel-Hagen.[6]
In 1939, Hausdorff asked the mathematician Richard Courant, in vain, for a research fellowship to be able to emigrate into the USA. In mid-1941, the Bonn Jews began to be deported to the "Monastery for Eternal Adoration" in
Some of his fellow Jews may have had illusions about the camp Endenich, but not Hausdorff. In the estate of Bessel-Hagen, E. Neuenschwander discovered the farewell letter that Hausdorff wrote to his lawyer Hans Wollstein, who was also Jewish.[8][9] Here is the beginning and end of the letter:
Dear friend Wollstein!
If you receive these lines, we (three) have solved the problem in a different manner — in the manner of which you have constantly tried to dissuade us. The feeling of security that you have predicted for us once we would overcome the difficulties of the move, is still eluding us; on the contrary, Endenich may not even be the end!
What has happened in recent months against the Jews evokes justified fear that they will not let us live to see a more bearable situation.
After thanking friends and, in great composure, expressing his last wishes regarding his funeral and his will, Hausdorff writes:
I am sorry that we cause you yet more effort beyond death, and I am convinced that you are doing what you can do (which perhaps is not very much). Forgive us our desertion! We wish you and all our friends to experience better times.
Your truly devoted
Felix Hausdorff
Unfortunately, this desire was not fulfilled. Hausdorff's lawyer, Wollstein, was murdered in
Hausdorff's library was sold by his son-in-law and sole heir, Arthur König. The portions of Hausdorff's Nachlass which could be saved by Hans Bonnet are now in the university and State Library of Bonn. The Nachlass is catalogued.[10]
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
witch trialsfade 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 in 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 in 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.
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 be placed in the sequence ? In a letter to Hilbert on 29 September 1904, he speaks of this problem, "it has plagued me almost like monomania".[11] Hausdorff saw a new strategy to attack the problem in the set . Cantor had suspected , but had only been able to show that . While is the "number" of possible
In 1904 Hausdorff published the recursion named after him, which states that for each non-limit ordinal we have
This formula was, together with a later notion called cofinality introduced by Hausdorff, the basis for all further results for
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. An ordinal is called regular if it is cofinal with any smaller ordinal; otherwise it is called singular. 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".[13]
The following theorem due to Hausdorff is also of fundamental importance: for each unbounded and ordered dense set there are two uniquely determined regular initial numbers so that is cofinal with and coinitial with (where * denotes the inverse order). This theorem provides, for example, a technique to characterize elements and gaps in ordered sets.
If is a predetermined set of characters (element and gap characters), the question arises whether there are ordered sets whose character set is exactly . One can easily find a necessary condition for , 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.[14] In this reservoir can be found interesting structures like the Hausdorff normal-types, in connection with which Hausdorff first formulated the
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.[16]
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.[17]
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, - and -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 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,
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
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
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
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
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.[23]
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
A significant contribution to the emerging field of functional analysis in the 1920s was Hausdorff's extension of the
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
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.[26]
His last works
In 1938, Hausdorff's last work Extension of a continuous map showed that a continuous function from a closed subset of a metric space can be extended to all of (although the image may need to be extended). As a special case, every homeomorphism from can be extended to a homeomorphism from . 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 be a metric space, a closed subset. If 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
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.[27]
The Asteroid
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.
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. Annual report of the DMV 16 (1907), S. 541–546.
- Grundzüge einer Theorie der geordneten Mengen. 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. Math. Annalen 77 (1916), S. 430–437.
- Dimension und äußeres Maß. Math. Annalen 79 (1919), S. 157–179.
- Über halbstetige Funktionen und deren Verallgemeinerung. Math. Zeitschrift 5 (1919), S. 292–309.
- Summationsmethoden und Momentfolgen I, II. 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.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,
- Band III: Mengenlehre (1927, 1935); Deskriptive Mengenlehre und Topologie. 2008, ISBN 978-3-540-76806-7
- Band IV: Analysis, Algebra und Zahlentheorie. 2001,
- Band V: Astronomie, Optik und Wahrscheinlichkeitstheorie. 2006,
- Band VI: Geometrie, Raum und Zeit. 2020. ISBN 978-3-540-77838-7
- Band VII: Philosophisches Werk. 2004,
- Band VIII: Literarisches Werk. 2010, ISBN 978-3-540-77758-8
- Band IX: Korrespondenz. 2012, ISBN 978-3-642-01116-0.
References
- Alexandroff, P.; Hopf, H.: Topologie. Springer-Verlag, Berlin 1935.
- Brieskorn, E.: Gustav Landauer und der Mathematiker Felix Hausdorff. In: H. Delf, G. Mattenklott: Gustav Landauer im Gespräch – Symposium zum 125. Geburtstag. Tübingen 1997, S. 105–128.
- Brieskorn, E. (Hrsg.): Felix Hausdorff zum Gedächtnis. Aspekte seines Werkes. Vieweg, Braunschweig/Wiesbaden 1996.
- Brieskorn, E.; Purkert, W.: Felix Hausdorff-Biographie. (Band IB der Edition), Springer, Heidelberg 2018.
- Eichhorn, E.; Thiele, E.-J.: Vorlesungen zum Gedenken an Felix Hausdorff, ISBN 3-88538-105-2.
- Koepke, P., Kanovei V., Deskriptive Mengenlehre in Hausdorffs Grundzügen der Mengenlehre, 2001, uni-bonn.de (pdf)
- Lorentz, G. G.: Das mathematische Werk von Felix Hausdorff. Jahresbericht der DMV 69 (1967), 54 (130)-62 (138).
- Purkert, Walter: The Double Life of Felix Hausdorff/Paul Mongré. Mathematical Intelligencer, 30 (2008), 4, S. 36 ff.
- Purkert, Walter: Felix Hausdorff - Paul Mongré. Mathematician - Philosopher - Man of Letters. Hausdorff Center for Mathematics, Bonn 2013.
- Stegmaier, W.: Ein Mathematiker in der Landschaft Zarathustras. Felix Hausdorff als Philosoph. Nietzsche-Studien 31 (2002), 195–240.
- Vollhardt, F.: Von der Sozialgeschichte zur Kulturwissenschaft? Die literarisch-essayistischen Schriften des Mathematikers Felix Hausdorff (1868–1942): Vorläufige Bemerkungen in systematischer Absicht. In: Huber, M.; Lauer, G. (Hrsg.): Nach der Sozialgeschichte - Konzepte für eine Literaturwissenschaft zwischen Historischer Anthropologie, Kulturgeschichte und Medientheorie. Max Niemeier Verlag, Tübingen 2000, S. 551–573.
- Wagon, S.: The Banach–Tarski Paradox. Cambridge Univ. Press, Cambridge 1993.
- Lexikon deutsch-jüdischer Autoren , Band 10, Saur, München 2002, S. 262–268
See also
- Gromov–Hausdorff convergence
- Hausdorff distance
- Hausdorff gap
- Maurice René Fréchet
- Hausdorff Medal
References
- ^ "Hausdorff space Definition & Meaning". Retrieved 15 June 2022.
- ^ Purkert, Prof. Dr. Walter. "Felix Hausdorff - Paul Mongré" (PDF). Hausdorff Edition. Retrieved November 14, 2023.
- .
- ^ Archiv der Universität Leipzig, PA 547
- ISBN 9780444516213.
- ^ Neuenschwander, E.: Felix Hausdorffs letzte Lebensjahre nach Dokumenten aus dem Bessel-Hagen-Nachlaß. In: Brieskorn 1996, S. 253–270.
- ^ Schubring, G. (2021). Biographie. Felix Hausdorff, Gesammelte Werke. Band IB. by Egbert Brieskorn and Walter Purkert. Math Intelligencer 43, 94–98
- ^ Nachlass Bessel-Hagen, Universitätsarchiv Bonn. Abgedruckt in Brieskorn 1996, S. 263–264 und im Faksimile S. 265–267
- ^ The full text of Abschiedsbrief Felix Hausdorffs at Wikisource
- ^ Siehe Findbuch Nachlass Hausdorff
- ^ Niedersächsische Staats- und Universitätsbibliothek zu Göttingen, Handschriftenabteilung, NL Hilbert, Nr. 136.
- ^ Detaillierte Angaben findet man in den gesammelten Werken, Band II, S. 9–12.
- ^ H.: Gesammelte Werke. Band II: Grundzüge der Mengenlehre. Springer-Verlag, Berlin, Heidelberg etc. 2002. Kommentare von U. Felgner, S. 598–601.
- ^ H.: Gesammelte Werke. Band II: Grundzüge der Mengenlehre. Springer-Verlag, Berlin, Heidelberg etc. 2002. S. 604–605.
- ^ Siehe dazu den Essay von U. Felgner: Die Hausdorffsche Theorie der -Mengen und ihre Wirkungsgeschichte in H.: Gesammelte Werke. Band II: Grundzüge der Mengenlehre. Springer-Verlag, Berlin, Heidelberg etc. 2002. S. 645–674.
- ^ Siehe dazu und zu ähnlichen Sätzen von Kuratowski und Zorn den Kommentar von U. Felgner in den gesammelten Werken, Band II, S. 602–604.
- ^ Schoenflies, A.: Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Teil II. Jahresbericht der DMV, 2. Ergänzungsband, Teubner, Leipzig 1908., S. 40.
- ^ For the history of Haussdorff's sphere paradox see Gesammelte Werke Band IV, S. 11–18; also the article by P. Schreiber in Brieskorn 1996, S. 135–148, and the monograph Wagon 1993.
- ^ Urysohn, P.: Mémoire sur les multiplicités Cantoriennes. (PDF; 6,2 MB) Fundamenta Mathematicae 7 (1925), S. 30–137; 8 (1926), S. 225–351.
- ^ P. Alexandroff: Sur la puissance des ensembles mesurables B. Comptes rendus Acad. Sci. Paris 162 (1916), S. 323–325.
- ^ W. H. Young: Zur Lehre der nicht abgeschlossenen Punktmengen. Berichte über die Verhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig, Math.-Phys. Klasse 55 (1903), S. 287–293.
- ^ Alexandorff, Hopf 1935, S. 20. For details see Gesammelte Werke Band II, S. 773–787.
- ^ For the history of the reception of Dimension und äußeres Maß, see the article by Bandt/Haase and Bothe/Schmeling in Brieskorn 1996, S. 149–183 and S. 229–252 and the commentary of S. D. Chatterji in Gesammelten Werken, Band IV, S. 44–54 and the literature given there.
- ^ Gesammelte Werke Band IV, S. 105–171, 191–235, 255–267 and 339–373.
- ^ See commentary by S. D. Chatterji in Gesammelten Werken Band IV, S. 182–190.
- Monatshefte für Mathematik und Physik. 35: 56–58.
- ^ Ratsversammlung vom 18. Mai 2011 (Beschluss-Nr. RBV-822/11), amtliche Bekanntmachung: Leipziger Amtsblatt Nr. 11 vom 4. Juni 2011, bestandskräftig seit dem 5. Juli 2011 bzw. 5. August 2011. Vgl. Leipziger Amtsblatt Nr. 16 vom 10. September 2011.
- ^ "Review von Jeremy Gray der Bände 1a, 3, 8, 9, Bulletin AMS, Band 51, 2014, 169–172".
- ^ (PDF) from the original on 2015-09-28.
External links
- Felix Hausdorff at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "Felix Hausdorff", MacTutor History of Mathematics Archive, University of St Andrews
- Homepage of the Hausdorff Edition (German)
- Hausdorff Findbuch
- Hausdorff Center for Mathematics in Bonn
- Memorial "Stolperstein" at Hausdorff's last home, in Bonn