Felix Hausdorff

Felix Hausdorff

Felix Hausdorff (born November 8, 1868 in Breslau ; died January 26, 1942 in Bonn ) was a German mathematician .

He is considered a co-founder of general topology and made significant contributions to general and descriptive set theory , measure theory , functional analysis and algebra . His last chair was in Bonn.

In addition to his job, he also worked as a philosophical writer and man of letters under the pseudonym Paul Mongré . He was persecuted by the National Socialists and committed suicide to escape the concentration camp system .

Life and work

Childhood and youth

Hausdorff's father, the Jewish merchant Louis Hausdorff (1843–1896), moved with his young family to Leipzig in the autumn of 1870 and over the years ran various companies on Leipziger Brühl , including a linen and cotton goods store . He was an educated man and had won the Morenu title at the age of 14 . There are several treatises from his pen, including a longer work on the Aramaic translations of the Bible from the perspective of Talmudic law.

Hausdorff's mother Hedwig (1848–1902), who is also called Johanna in various documents, came from the extensive Jewish Tietz family. Hermann Tietz , founder of the first department store and later co-owner of the department store chain "Hermann Tietz" , also emerged from a branch of this family . During the time of the National Socialist dictatorship this was "Aryanized" under the name Hertie .

From 1878 onwards Felix Hausdorff attended the Nicolai-Gymnasium in Leipzig, an institution that had an excellent reputation as a nursery of humanistic education. He was an excellent student, top of his class for years, and was often honored by the fact that he was allowed to recite his own Latin or German poems at school celebrations. In his Abitur class of 1887 (with two top prima ) he was the only one who achieved the overall grade "I".

The choice of subject was not easy for Hausdorff. Magda Dierkesmann, who was a frequent guest at Hausdorff's home as a student in Bonn from 1926 to 1932, reported in 1967:

"His versatile musical talent was so great that it was only at the urging of his father that he gave up the plan to study music and become a composer."

For the Abitur , the decision was made in favor of the natural sciences.

Studies, doctorate and habilitation

From the summer semester 1887 to the summer semester 1891 Hausdorff studied mathematics and astronomy , mainly in his hometown Leipzig, interrupted by one semester each in Freiburg im Breisgau (summer semester 1888) and Berlin (winter semester 1888/1889). The surviving study reports show him as an extraordinarily versatile young man who, in addition to the mathematical and astronomical lectures, also attended lectures in physics, chemistry and geography, as well as lectures on philosophy and the history of philosophy as well as on topics in linguistics, literature and social sciences. In Leipzig he heard the musicologist Paul’s lecture on the history of music. His early love for music lasted a lifetime; In Hausdorff's house there were impressive music evenings with the host at the piano, as testimony of various participants. Even as a Leipzig student, he was an admirer and connoisseur of Richard Wagner's music .

In the last semesters of his studies, Hausdorff closely followed Heinrich Bruns (1848–1919). Bruns was professor for astronomy and director of the observatory at the University of Leipzig. With him Hausdorff received his doctorate in 1891 with the work on the theory of astronomical ray refraction on the refraction of light in the atmosphere. This was followed by two further publications on the same topic and in 1895 the habilitation with a thesis on the extinction of light in the atmosphere. These early astronomical works by Hausdorff - regardless of their excellent mathematical work-through - gained no significance. On the one hand, the underlying idea by Bruns has proven to be unsustainable (near-horizon astronomical refraction observations were required, which - as Julius Bauschinger was able to show a little later - in principle cannot be obtained with the required accuracy). On the other hand, the progress made in direct measurement of atmospheric data (balloon ascents) soon made the laborious calculation of these data from refraction observations unnecessary. In the period between doctorate and habilitation , Hausdorff completed the one-year voluntary military service and worked as a computer at the Leipzig observatory for two years .

Private lecturer in Leipzig

With his habilitation, Hausdorff became a private lecturer at the University of Leipzig and began extensive teaching activities in a wide variety of mathematical fields. In addition to teaching and researching mathematics, he pursued his literary and philosophical inclinations. As a man with diverse interests, comprehensively educated, highly sensitive and differentiated in thinking, feeling and experiencing, he associated with a number of well-known writers, artists and publishers such as Hermann Conradi , Richard Dehmel , Otto Erich Hartleben , Gustav Kirstein and Max Klinger during his time in Leipzig , Max Reger and Frank Wedekind . The years 1897 to around 1904 mark the climax of his literary and philosophical work; During this time, 18 of the total of 22 published under a pseudonym, including a volume of poetry, a play, an epistemological book and a volume of aphorisms appeared .

Hausdorff married Charlotte Goldschmidt in 1899, the daughter of the Jewish doctor Siegismund Goldschmidt from Bad Reichenhall. His stepmother was the famous women's rights activist and preschool teacher Henriette Goldschmidt . Hausdorff's only child, the daughter Lenore (Nora), was born in 1900; she survived the Nazi era and died very old in 1991 in Bonn.

Places of activity as a professor

In December 1901 Hausdorff was appointed adjunct professor at the University of Leipzig. The often repeated claim that Hausdorff received a call from Göttingen and rejected it cannot be archived and is probably wrong. When applying in Leipzig, the dean Kirchner felt compelled to add the following addition to the very positive opinion of the specialist colleagues, written by Heinrich Bruns:

“However, the faculty considers itself obliged to report to the Royal Ministry that the above application will be submitted in the November 2nd d. J. was not accepted with all, but with 22 against 7 votes. The minority voted against it because Dr. Hausdorff is of the Mosaic faith. "

This addition highlights the undisguised anti-Semitism , which especially after the founder crash of 1873 had a strong upswing in the entire German Empire. Leipzig was a center of the anti-Semitic movement, especially among the student body. This may have been one reason why Hausdorff did not feel particularly comfortable at Leipzig University; Another was perhaps the emphatically hierarchical behavior of the Leipzig Ordinaries, where the Extraordinarius was of no importance.

After completing his habilitation, Hausdorff wrote another thesis on optics, one on non-Euclidean geometry and one on hyper-complex number systems, as well as two papers on probability theory . However, his main area of ​​work soon became set theory, especially the theory of ordered sets. It was initially a philosophical interest that led him to study Georg Cantor's work around 1897 . Hausdorff gave a lecture on set theory as early as the summer semester of 1901. This was one of the first lectures on set theory, only Ernst Zermelo's college in Göttingen in the winter semester 1900/1901 was a little earlier. Cantor himself never read about set theory. Hausdorff's first set-theoretical discovery can be found in this lecture: The type class of all countable order types has the strength of the continuum . However, this sentence was already found in Felix Bernstein's dissertation.

In the summer semester of 1910, Hausdorff was appointed as a scheduled associate professor at the University of Bonn. In Bonn he began with a lecture on set theory, which he repeated in the summer semester of 1912, substantially revised and expanded.

In the summer of 1912, work began on his magnum opus, the book Basic Principles of Set Theory, which was published in April 1914.

Hausdorff was appointed full professor at the University of Greifswald in the summer semester of 1913 . This university was the smallest among the Prussian universities. The mathematics institute was also small; in the summer semester of 1916 and in the winter semester of 1916/17 Hausdorff was the only mathematician in Greifswald. This meant that his teaching was almost completely occupied by the basic lectures.

The fact that Hausdorff was appointed to Bonn in 1921 meant a significant improvement in his scientific situation. Here he was able to develop a wide-ranging teaching activity and repeatedly present the latest research. Particularly noteworthy, for example, is a lecture on probability theory (NL Hausdorff: Kapsel 21: Fasz. 64.) from the summer semester of 1923, in which he established this theory axiomatically and in terms of mass theory, and this ten years before A. N. Kolmogoroff's Basic Concepts of Probability Calculus (fully printed in Works, Volume V). In Bonn Hausdorff had outstanding mathematicians as colleagues and friends with Eduard Study and later with Otto Toeplitz .

Hausdorff under the National Socialist dictatorship

Anti-Semitism became state doctrine when the National Socialists came to power . Hausdorff was initially not directly affected by the “ Law for the Restoration of the Professional Civil Service ” passed in 1933 , as he was a German civil servant before 1914. However, he was probably not spared the fact that one of his lectures was disrupted by National Socialist student officials. So he broke off his lecture on calculus III from the winter semester 1934/35 on November 20. Since a working conference of the National Socialist German Student Union (NSDStB) took place at Bonn University these days , which stipulated that the focus of the work in the current semester was the topic of "Race and Ethnicity", the assumption is very reasonable that Hausdorff abandoned the lecture has to do with this event, because he has never broken off a lecture in his long career as a university lecturer.

On March 31, 1935, Hausdorff finally retired after some back and forth. Those responsible at the time did not find a word of thanks for 40 years of successful work in German higher education. He worked tirelessly and, in addition to the expanded new edition of his set theory , published seven papers on topology and descriptive set theory, all of which appeared in Polish journals: one in Studia Mathematica , the rest in Fundamenta Mathematicae .

Hausdorff's estate also shows that in the increasingly difficult times he worked constantly mathematically and tried to follow current developments in the areas of interest to him. Erich Bessel-Hagen supported him selflessly by not only remaining loyal to the Hausdorff family in friendship, but also getting books and magazines from the institute library, which Hausdorff was no longer allowed to enter as a Jew.

A lot is known from various sources about the humiliations that Hausdorff and his family were exposed to, especially after the November pogroms in 1938 , e.g. B. from the letters of Bessel-Hagen.

In 1939 Hausdorff tried in vain to get a research fellowship in the USA through mathematician Richard Courant in order to be able to emigrate.

The first page of the farewell letter to Hans Wollstein.

In mid-1941, the Bonn Jews finally began to be deported to the “For Eternal Adoration” monastery in Bonn-Endenich , from which the nuns had been expelled. From there the transports to the extermination camps in the east followed. After Felix Hausdorff, his wife and his wife's sister, Edith Pappenheim, who lived with them, received the order in January 1942 to move to the Endeich camp, they passed away on January 26, 1942 by taking an overdose of veronal . Her final resting place is in the Poppelsdorf cemetery in Bonn. Between the order in the interim storage facility and the suicide, he gave his handwritten estate to the Egyptologist and presbyter Hans Bonnet , who was largely able to save him after his house was destroyed by a bomb.

Some of his Jewish fellow citizens may still have illusions about the Endeich camp - Hausdorff himself did not. E. Neuenschwander also discovered the farewell letter in the Bessel-Hagen estate that Hausdorff had written to the Jewish lawyer Hans Wollstein; here beginning and end of the letter:

Felix Hausdorff's grave in Bonn-Poppelsdorf

“Dear friend Wollstein!
If you receive these lines, the three of us have solved the problem in a different way - the way you have consistently tried to dissuade us. The feeling of security that you have predicted for us once we have overcome the difficulties of the move does not want to set in at all, on the contrary:
even Endeich
Perhaps the end is not!
What has happened against the Jews in the last few months arouses well-founded fear that we will no longer be allowed to experience a state that is tolerable for us. "

After thanking friends and after expressing his last wishes regarding the funeral and will in great composure, Hausdorff continues:

“Forgive us for causing you trouble after death; I am convinced that you do what you do can (and perhaps not very much). Also forgive us for our desertion ! We wish you and all of our friends to have even better times.
Your faithful
Felix Hausdorff "

Hausdorff's last written wish was not fulfilled: lawyer Wollstein was murdered in Auschwitz .

Hausdorffstrasse (Bonn)

Hausdorff's library was sold by his son-in-law and sole heir Arthur König. The handwritten estate was taken over for safekeeping by a family friend, the Bonn Egyptologist Hans Bonnet. Today it is in the University and State Library of Bonn . The estate is cataloged.

Work and reception

Hausdorff as a philosopher and writer (Paul Mongré)

His volume of aphorisms from 1897 was the first work by Hausdorff to appear under the pseudonym Paul Mongré ( à mon gré means: according to my wishes, as I like it). It bears the title of Sant 'Ilario. Thoughts from the landscape of Zoroaster. The subtitle of Sant 'Ilario “Thoughts from the landscape of Zarathustra” alludes to the fact that Hausdorff completed his book while relaxing on the Ligurian coast around Genoa and that Friedrich Nietzsche wrote the first two parts of Also sprach Zarathustra in this area ; he also alludes to the spiritual closeness to Nietzsche. In a self-advertisement by Sant 'Ilario in the weekly Die Zukunft , Hausdorff expressly acknowledged Nietzsche.

Hausdorff did not try to copy Nietzsche or even to surpass it. “No trace of Nietzsche imitation”, it says in a contemporary review. Alongside Nietzsche, he strives to set free individual thinking, to take the freedom to question traditional norms. Hausdorff kept a critical distance from Nietzsche's late work. In his essay on the book Der Wille zur Macht (The Will to Power) , compiled from Nietzsche 's notes left behind by the Nietzsche Archive , it says:

“A fanatic glows in Nietzsche. His moral of breeding, built on our present-day foundations of biological and physiological knowledge: that could become a world-historical scandal, against which the Inquisition and the witch trial fade into harmless aberrations. "

Hausdorff took his critical yardstick from Nietzsche himself,

"From the kind, measured, understanding free spirit Nietzsche and from the cool, dogma-free, systemless skeptic Nietzsche [...]"

In 1898 - also under the pseudonym Paul Mongré - Hausdorff's epistemological attempt The Chaos in Cosmic Selection was published. The metaphysics criticism presented in this book had its starting point in Hausdorff's analysis of Nietzsche's idea of ​​the eternal return . After all, the point is to finally destroy any kind of metaphysics. We know nothing and we cannot know anything about the world itself , about the transcendent core of the world - as Hausdorff puts it. We have to presuppose “the world in itself” as indefinite and indefinable, as mere chaos. The world of our experience, our cosmos, is the result of the selection, the selection that we have always involuntarily made and continue to make according to our possibilities of knowledge. Viewed from this chaos, any other order, other cosmoi, would also be conceivable. In any case, one cannot draw any conclusion from the world of our cosmos to a transcendent world.

In 1904, Hausdorff's play, the one-act play The Doctor of His Honor , appeared in the magazine Die neue Rundschau . It is a crude satire on the mischief of duels and on the traditional notions of honor of the nobility and the Prussian officer corps, which became more and more anachronistic in the developing bourgeois society. The doctor of his honor was Hausdorff's greatest literary success. There were numerous performances in more than thirty cities between 1904 and 1918. Hausdorff later wrote an epilogue to the play, but it was not performed at the time. It was not until 2006 that this epilogue was premiered at the annual meeting of the German Mathematicians Association in Bonn.

In addition to the works mentioned above, Hausdorff wrote numerous essays that appeared in leading literary magazines of the time, as well as a volume of poems Ekstasen (1900). Some of his poems were set to music by the Austrian composer Joseph Marx .

Ordered Set Theory

Hausdorff's entry into a thorough study of ordered sets was motivated not least by Cantor's continuum problem , which place the cardinal number takes in the series . In a letter to Hilbert on September 29, 1904, he said that this problem had " plagued him almost like a monomania ". He saw in the sentence a new strategy for tackling the problem. Cantor suspected; was only proven . is the "number" of possible well-orders of a countable set ; had now turned out to be the “number” of all possible orders of such a set. It was therefore natural to study orders that are more specific than any order, but more general than well-ordered orders. This is exactly what Hausdorff did in his first set-theoretical publication in 1901 with the study of “graduated sets”. It is known from the results of Kurt Gödel and Paul Cohen that this strategy of solving the continuum problem was just as unsuccessful as Cantor's strategy, which aimed to apply Cantor-Bendixson's theorem from closed sets to any uncountable point sets generalize. ${\displaystyle \aleph =2^{\aleph _{0}}}$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \mathrm {card} (T(\aleph _{0}))=\aleph }$ ${\displaystyle \aleph =\aleph _{1}}$ ${\displaystyle \aleph \geq \aleph _{1}}$ ${\displaystyle \aleph _{1}}$ ${\displaystyle \aleph }$

In 1904 Hausdorff published the recursion formula named after him :

The following applies to every non-limit number ${\displaystyle \mu }$ ${\displaystyle \aleph _{\mu }^{\aleph _{\alpha }}=\aleph _{\mu }\;\aleph _{\mu -1}^{\aleph _{\alpha }}.}$

This formula, together with the concept of confinality later introduced by Hausdorff , became the basis of all further results on Alep exposure . Exact knowledge of the problem of recursion formulas of this type enabled Hausdorff to  uncover the error in Julius Koenig's lecture at the 1904 International Congress of Mathematicians in Heidelberg. König had stated there that the continuum could not be well ordered, that is, that its cardinal number was not an aleph at all; he had caused a sensation with it. The statement that it was Hausdorff who cleared up the error is particularly significant, because historical literature has painted a wrong picture of the events in Heidelberg for more than 50 years.

In the years 1906 to 1909 Hausdorff's fundamental works on ordered quantities fall. Only a few points can be briefly touched on here. The concept of confinality introduced by Hausdorff is of fundamental importance for the entire theory. An ordinal number is called regular if it is not confinal with any smaller ordinal number, otherwise singular. Hausdorff's question as to whether there are regular initial numbers with a limit index was the starting point for the theory of unreachable cardinal numbers. Hausdorff had already noticed that such numbers, if they exist, must be of "exorbitant size".

The following theorem of Hausdorff is of fundamental importance: For every ordered unrestricted dense set there are two uniquely determined regular initial numbers such that with confinal, with (* denotes the inverse order) is coinitial. For example, this theorem provides a technique to characterize gaps and elements in ordered sets. Hausdorff used the gap and elementary characters that he had introduced. ${\displaystyle A}$ ${\displaystyle \omega _{\xi },\omega _{\eta }}$ ${\displaystyle A}$ ${\displaystyle \omega _{\xi }}$ ${\displaystyle \omega _{\eta }^{*}}$

If there is a given set of characters (element and gap characters), the question arises as to whether there are ordered sets whose character set is even . It is relatively easy to find a necessary condition for . Hausdorff succeeded in showing that this condition is also sufficient, i. that is, for everyone that satisfies the condition, there is an ordered set that has the character set. For this one needs a rich reservoir of ordered quantities; Hausdorff was also able to achieve this with his theory of generally ordered products and potencies. Interesting structures such as Hausdorff's normal types can be found in this reservoir ; In connection with their studies, Hausdorff first formulated the generalized continuum hypothesis. Hausdorff's quantities formed the starting point for the study of the saturated structures that are so important in model theory . ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle \eta _{\alpha }}$ ${\displaystyle \eta _{\alpha }}$

Hausdorff's general products and potencies had also led him to the concept of the partially ordered set. Furthermore, the final gradations of sequences or functions, which he studied in detail, turned out to be partial orders. The problem of whether there are maximally ordered subsets (Hausdorff called them pantachia) without gaps in these partially ordered sets is the oldest problem in set theory that has not yet been solved. The question whether there is a maximum ordered subset containing each ordered subset of a partially ordered set was answered positively by Hausdorff using the well-ordered theorem. This is the maximum chain set named after him today . It does not only follow from the well-ordered theorem (or from the axiom of choice, which is equivalent to it), but it is, as it turned out later, even equivalent to the axiom of choice. ${\displaystyle \omega _{1}\omega _{1}^{*}}$

As early as 1908, Arthur Moritz Schoenflies had stated in the second part of his report on set theory that the more recent theory of ordered sets (i.e. the extensions of this theory according to Cantor) were almost entirely due to Hausdorff.

The opus magnum "Principles of Set Theory"

Set theory in the understanding of this area at the time included not only general set theory but also the theory of point sets as well as content and measure theory. Hausdorff's work was the first textbook that presented the entire set of theory in this comprehensive sense systematically and with complete evidence. Hausdorff was aware of how easily the human mind can err in striving for rigor and truth. So he promises in the foreword to the main features :

"... to use the human privilege of error as sparingly as possible."

This book went far beyond the masterful portrayal of the familiar. It contained a number of significant original contributions by its author, which can only be outlined briefly below.

The first six chapters of the Fundamentals deal with general set theory. At the top, Hausdorff puts a detailed set algebra with partly new, forward-looking concepts (difference chains, quantity rings and quantity bodies and systems). These introductory paragraphs on sets and their connections also contain, for example, the modern set-theoretical function concept ; they provide the mathematical language of the future, so to speak. Chapters 3 to 5 are followed by the classical theory of cardinal numbers, order types and ordinal numbers . In the sixth chapter "Relationships between ordered and well-ordered sets", Hausdorff presents, among other things, the most important results of his own research on ordered sets. ${\displaystyle \delta }$ ${\displaystyle \sigma }$

In the chapters on "point sets" - the topological chapters - Hausdorff first developed a systematic theory of topological spaces based on his well-known environmental axioms , whereby he also called for the axiom of separation that was later named after him . This theory emerges from a comprehensive synthesis of earlier approaches by other mathematicians and Hausdorff's own reflections on the spatial problem. The concepts and propositions of the classical point set theory are - as far as possible - transferred to the general case and thus part of the newly created general or set theoretical topology. But Hausdorff not only does this "translation work", he also develops fundamental construction methods of the topology such as kernel formation (open core, self-contained core ) and shell formation ( closed shell ), and he works on the fundamental meaning of the concept of the open set (from him, " Area ”) and the compactness concept introduced by Fréchet . He also justifies and develops the theory of the connection , in particular by introducing the terms “component” and “quasi-component”. ${\displaystyle \mathbb {R} ^{n}}$

Using the first and finally the second Hausdorff countability axiom , the considered spaces are gradually further specialized. A large class of spaces that satisfy the first axiom of countability are the metric spaces . They were introduced in 1906 by Fréchet under the name "classes (E)". Hausdorff gave the name "metric space". He systematically developed the main features of the theory of metric spaces and enriched it with a number of new concepts: Hausdorff metrics, completion, total limitation, context, reducible sets. Fréchet's work had received little attention; Only through Hausdorff's basic features did metric spaces become common property of mathematicians. ${\displaystyle \rho }$

The chapter on illustrations and the final chapter on the fundamentals of measurement and integration theory are also impressive due to the generality of the position taken and the originality of the presentation. Hausdorff's reference to the importance of the theory of measurements for the calculation of probability had - although laconic brevity - had a great historical impact. This chapter also contains the first correct proof of Émile Borel's strong law of large numbers . Finally, the appendix contains what is probably the most spectacular individual result of the entire book, namely Hausdorff's proposition that one can define a content for subsets that are not restricted to all. The proof is based on Hausdorff's paradoxical decomposition of spheres, for the production of which one needs the axiom of choice. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n\geq 3}$

In the course of the 20th century it became the standard to build mathematical theories axiomatically using set theory. The creation of axiomatically based general theories, such as general topology, served, among other things, to peel out the common structural core from various concrete cases or sub-areas and then to set up an abstract theory which contained all these parts as special cases and which offered great profit Simplification, standardization and thus ultimately an economy of thought with it. Hausdorff even this aspect has the main features highlighted. Seen in this way, the topological chapters of the basic features are methodologically a pioneering achievement, and in this respect they pointed the way for the development of modern mathematics.

The basics of set theory appeared in an already tense time on the eve of the First World War . In August 1914 the war began, which also dramatically affected scientific life in Europe. Under these circumstances, Hausdorff's book could hardly take effect in the first five to six years after its publication. After the war, a young new generation of researchers set out to take up the suggestions so abundantly contained in this work, undoubtedly focusing on the topology. 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 specialist journals with a focus on set theory, topology, theory of real functions, measurement and integration theory, functional analysis, logic and the fundamentals of mathematics. The general topology was of particular importance in this spectrum. Hausdorff's main features were present in Fundamenta Mathematicae with remarkable frequency from the first volume. Of the 558 works (not counting Hausdorff's own three works) that appeared in the first twenty volumes from 1920 to 1933, 88 cite the main features. One must also take into account that Hausdorff's conceptual formations increasingly became common knowledge, so that they are also used in a number of works that do not explicitly name him.

The Russian topological school, founded by Paul Alexandroff and Paul Urysohn , was also based to a large extent on Hausdorff's main features. The correspondence with Urysohn and in particular Alexandroff and also Urysohn's Mémoire sur les multiplicités Cantoriennes , a work the size of a book in which Urysohn develops his theory of dimensions and in which the main features are cited no less than 60 times, testifies to this.

Hausdorff's book was in brisk demand long after World War II, and Chelsea had three reprints in 1949, 1965 and 1978.

Descriptive set theory, measure theory and analysis

In 1916 Hausdorff and Alexandroff solved the continuum problem for Borel sets independently of one another : every Borel set in a complete separable metric space is either at most countable or it has the thickness of the continuum. This result generalizes the Cantor-Bendixson theorem, which makes such a statement for closed sets of . For linear quantities had William Henry Young in 1903, for quantities Hausdorff 1914 in the fundamentals achieved an appropriate result. The theorem of Alexandroff and Hausdorff was a powerful impetus for the further development of descriptive set theory. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle G_{\delta }}$ ${\displaystyle G_{\delta \sigma \delta }}$

From Hausdorff's publications during the Greifswald period, the work from 1919 Dimension and external dimensions stand out in particular. It has remained highly topical to this day and is probably the most cited original mathematical work from the decade from 1910 to 1920 in recent years. This work introduces the concepts that are now known as the Hausdorff dimension and the Hausdorff dimension .

Hausdorff's concept of dimensions is a fine instrument for characterizing and comparing “strongly fissured sets”. The concepts of dimension and external measure have been used and further developed in numerous areas, such as in the theory of dynamic systems, geometric measure theory, the theory of self-similar sets and fractals , the theory of stochastic processes, harmonic analysis, potential theory and number theory.

Important analytical work by Hausdorff fell into his second period in Bonn. In Summation Methods and Moment Sequences I , in 1921, he developed a whole class of summation methods for divergent series , which are now called Hausdorff methods . In Hardy's classic Divergent Series , a whole chapter is devoted to the Hausdorff process. The classic processes of Hölder and Cesàro turned out to be special Hausdorff processes. Every Hausdorff method is given by a sequence of moments; In this context Hausdorff gave an elegant solution to the moment problem for a finite interval, bypassing the theory of continued fractions. In moment problems for a finite interval of 1923 he dealt with more specific moment problems, for example with certain restrictions on the generating density , e.g. B. . Criteria for solvability and definiteness of momentary problems have occupied Hausdorff for many years, as hundreds of pages of studies in his estate attest. ${\displaystyle \varphi (x)}$ ${\displaystyle \varphi (x)\in L^{p}[0,1]}$

An important contribution to the functional analysis that developed in the 1920s was Hausdorff's translation of Fischer-Riesz's theorem on -Räume in 1923 into an extension of Parseval's theorem on Fourier series. There he proved the inequalities named after him and W. H. Young today. The Hausdorff-Young inequalities have become the starting point for far-reaching new developments. ${\displaystyle L^{p}}$

In 1927 Hausdorff's book set theory was published. It was declared as the 2nd edition of the basics , but actually a completely new book. Since the scope was considerably limited compared to the basic features due to the appearance in Göschen's teaching library , large parts of the theory of ordered sets and the theory of measure and integration were omitted. “More than these deletions will perhaps be regretted” (so Hausdorff in the foreword) “that I have given up the topological point of view, through which the first edition apparently made many friends, in order to save further space in point set theory, and instead rely on the simpler theory of metric spaces ”.

In fact, some reviewers of the work have expressly regretted this. As a sort of compensation, Hausdorff presented the then current state of descriptive set theory for the first time. This fact ensured that the book received almost as intense a reception as the basic features , especially in Fundamenta Mathematicae. It was very popular as a textbook; In 1935 an expanded new edition appeared; this was reprinted at Dover in 1944. An English translation appeared in 1957, with reprints in 1962 and 1967.

There is also a Russian edition (1937), which, however, is only partly a faithful translation, partly a revision by Alexandroff and Kolmogorow , which brought the topological point of view back to the fore. In 1928 a review of set theory was published by Hans Hahn . Hahn may already have had the danger of German anti-Semitism in mind when he closed this meeting with the following sentence:

“An exemplary representation of a difficult and thorny area in every respect; a work of the kind who have carried the fame of German science across the world and of which all German mathematicians can be proud of its author. "

The last work

In his last work, Expansion of a Continuous Mapping , Hausdorff showed in 1938 that a continuous mapping can be continuously expanded from a closed subset of a metric space to the whole (the image space may have to be expanded). In particular, each homeomorphism can be expanded from to a homeomorphism to entirely . This work continues research from previous years. In 1919 Hausdorff had given, among other things, a new simple proof of Tietze's continuation in About semi-continuous functions and their generalization . In 1930 he showed the following in an extension of a homeomorphism : If a metric space is closed and a new metric is introduced without changing the topology, the new metric can be extended to the whole space while maintaining the old topology. The work Stepped Spaces was published in 1935, here Hausdorff considered spaces that satisfy the Kuratovskian shell axioms except for the axiom of the idempotency of the shell operator . He calls them stepped spaces (today often referred to as closure spaces ) and uses them to study the relationships between the Fréchetian Limes spaces and the topological spaces . The most important work of the 1930s is Sums of Quantities. It found an extraordinary response in the set theory of the “forcing era” (keyword “ Hausdorff gaps ”). ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle F\subset E}$ ${\displaystyle F}$ ${\displaystyle \aleph _{1}}$

Hausdorff as the namesake

The name Hausdorff is often found in mathematics, among other things the following were named after him:

• Hausdorff room
• Hausdorff measure
• Hausdorff dimension
• Hausdorff convergence
• Hausdorff metric
• Hausdorff's maximum chain sentence
• Hausdorff separation axiom
• Hausdorff gaps
• Hausdorff-Young inequality
• Baker-Campbell-Hausdorff formula
• Hausdorff's Gδ theorem

At the universities of Bonn and Greifswald he was named in his honor:

• which was founded in late 2006 Excellence Cluster Hausdorff Center for Mathematics in Bonn,
• the Hausdorff Research Institute for Mathematics in Bonn and that
• international meeting center Felix Hausdorff in Greifswald.

There is also Hausdorffstraße in Bonn, where he once lived (house number 61). In Greifswald there is a Felix-Hausdorff-Straße, which is where the institutes for biochemistry and physics are located. The newly created Hausdorffweg has been in the Leipzig district of Gohlis- Mitte since 2011 .

The asteroid (24947) Hausdorff was named after him.

Fonts

As Paul Mongré

Only a selection of the essays mentioned in the text is given here.

• Sant 'Ilario. Thoughts from the landscape of Zoroaster. Publisher CG Naumann, Leipzig 1897.
• The chaos in cosmic selection - an epistemological attempt. Verlag CG Naumann, Leipzig 1898; Reprint, ed. and with a preface by Max Bense. Agis-Verlag, Baden-Baden 1976, ISBN 3-87007-013-7 .
• Mass happiness and individual happiness. Neue Deutsche Rundschau (Free Stage) 9 (1), (1898), pp. 64–75.
• The unclean century. Neue Deutsche Rundschau (Free Stage) 9 (5), (1898), pp. 443–452.
• Ecstasies. Poetry book. Publisher H. Seemann Nachf., Leipzig 1900.
• The will to power. In: Neue Deutsche Rundschau (Free Stage) 13 (12) (1902), pp. 1334–1338.
• Max Klinger's Beethoven. Journal of Fine Arts, New Series 13 (1902), pp. 183–189.
• Language criticism. Neue Deutsche Rundschau (Free Stage) 14 (12), (1903), pp. 1233–1258.
• The doctor of his honor, grotesque. In: Die neue Rundschau (Free Stage) 15 (8), (1904), pp. 989-1013. New edition as: The doctor of his honor. Comedy in one act with an epilogue. With 7 portraits, woodcuts by Hans Alexander Müller after drawings by Walter Tiemann, 10 sheets, 71 p. Fifth regular publication of the Leipziger Bibliophilen-Evenings, Leipzig 1910. Reprint: S. Fischer, Berlin 1912, 88 p.

As Felix Hausdorff

• Contributions to the calculation of probability Ber. on the negotiations of the Royal. Saxon. Ges. The Wiss. to Leipzig. Math. Phys. Classe 53 (1901), pp. 152-178.
• About a certain kind of ordered set. Ber. on the negotiations of the Royal. Saxon. Ges. The Wiss. to Leipzig. Math. Phys. Classe 53 (1901), pp. 460-475.
• The space problem. Inaugural lecture at the University of Leipzig, held on July 4, 1903. Ostwalds Annalen der Naturphilosophie 3 (1903), pp. 1–23.
  • Also in: Herbert Beckert, Walter Purkert: Leipzig Mathematical Inaugural Lectures. Selection from the years 1869–1922. BG Teubner, Leipzig 1987 (with biography).
• The power concept in set theory. Annual report of DMV 13 (1904), pp. 569–571.
• Investigations into order types I, II, III. Ber. on the negotiations of the Royal. Saxon. Ges. The Wiss. to Leipzig. Math.-phys. \ Klasse 58 (1906), pp. 106-169.
• Studies on order types IV, V. Ber. on the negotiations of the Royal. Saxon. Ges. The Wiss. to Leipzig. Math. Phys. Class 59 (1907), pp. 84-159.
• About dense order types. Annual report of DMV 16 (1907), pp. 541–546.
• Outlines of a theory of ordered sets. Math. Annalen 65 (1908), pp. 435-505.
• The graduation after the final course. Treatises of the Royal. Saxon. Ges. The Wiss. to Leipzig. Math. Phys. Class 31 (1909), pp. 295-334.
• Basics of set theory . Publisher Veit & Co, Leipzig. 476 p. With 53 figures. Reprints: Chelsea Pub. Co., 1949, 1965, 1978.
• The power of the Borel sets. Math. Annalen 77 (1916), pp. 430-437.
• Dimension and external measure. Math. Annalen 79 (1919), pp. 157-179.
• About semi-continuous functions and their generalization. Math. Zeitschrift 5 (1919), pp. 292-309.
• Summation methods and momentary sequences I , II. Math. Zeitschrift 9 (1921), I: pp. 74-109, II: pp. 280-299.
• An extension of Parseval's theorem over Fourier series. Math. Zeitschrift 16 (1923), pp. 163-169.
• Moment problems for a finite interval. Math. Zeitschrift 16 (1923), pp. 220-248.
• Set theory. Second revised edition. Verlag Walter de Gruyter & Co., Berlin. 285 p. With 12 figures.
• Extension of a homeomorphy. (PDF; 389 kB), Fundamenta Mathematicae 16 (1930), pp. 353-360.
• On the theory of linear metric spaces. Journal for pure and applied mathematics 167 (1931/32), pp. 294-311.
• About inner pictures. Fundamenta Mathematicae 23 (1934), pp. 279-291.
• Set theory. Third edition. With an additional chapter and a few addenda. Verlag Walter de Gruyter & Co., Berlin. 307 p. With 12 figures. Reprint: Dover Pub. New York, 1944. English edition: Set theory. Translation from German by J. R. Aumann et al., Chelsea Pub. Co., New York 1957, 1962, 1967.
• Tiered rooms. (PDF; 1.2 MB), Fundamenta Mathematicae 25 (1935), pp. 486–502.
• Sums of amounts . ${\displaystyle \aleph _{1}}$Fundamenta Mathematicae 26 (1936), pp. 241-255.
• Expansion of a continuous map. (PDF; 450 kB), Fundamenta Mathematicae 30 (1938), pp. 40–47.
• Legacy writings. 2 volumes. Ed .: G. Bergmann, Teubner, Stuttgart 1969. Volume I contains the fascicles 510-543, 545-559, 561-577, Volume II the fascicles 578-584, 598-658 (all fascicles are reproduced in facsimile ).

Collected Works

The project "Hausdorff Edition" ( E. Brieskorn  (†), F. Hirzebruch  (†), W. Purkert , R. Remmert  (†) and E. Scholz ) has a commented on with authors from Germany and four other countries Estate material supplemented edition of the collected works started and largely completed. More than twenty mathematicians, historians, philosophers and literary scholars worked together. The edition was carried out as a long-term project by the North Rhine-Westphalian Academy of Sciences and Arts until the end of 2011 and funded as part of the academy program. The volumes are published by Springer-Verlag , Heidelberg; Nine volumes are planned, of which Volume I will be in two sub-volumes. All volumes except Volume VI have been published by 2018; the outstanding volume is expected to be released in 2019; This concludes the Hausdorff Edition.

• Volume IA: General set theory. 2013, ISBN 978-3-642-25598-4 .
• Volume IB: Felix Hausdorff - Paul Mongré (biography). 2018, ISBN 978-3-662-56380-9 .
• Volume II: Principles of Set Theory (1914). 2002, ISBN 978-3-540-42224-2 .
• Volume III: Set theory (1927, 1935); Descriptive set theory and topology. 2008, ISBN 978-3-540-76806-7 .
• Volume IV: Analysis, Algebra and Number Theory. 2001, ISBN 978-3-540-41760-6 .
• Volume V: Astronomy, Optics and Probability Theory. 2006, ISBN 978-3-540-30624-5 .
• Volume VI: Geometry, Space and Time.
• Volume VII: Philosophical Work. 2004, ISBN 978-3-540-20836-5 .
• Volume VIII: Literary Work. 2010, ISBN 978-3-540-77758-8 .
• Volume IX: Correspondence. 2012, ISBN 978-3-642-01116-0 .

literature

• Pavel Alexandroff , Heinz Hopf : Topology. Springer-Verlag, Berlin 1935.
• Egbert Brieskorn : Gustav Landauer and the mathematician Felix Hausdorff. In: H. Delf, G. Mattenklott: Gustav Landauer in conversation - Symposium for the 125th birthday. Tübingen 1997, pp. 105-128.
• Egbert Brieskorn (ed.): Felix Hausdorff for memory. Aspects of his work. Vieweg, Braunschweig / Wiesbaden 1996.
• Egbert Brieskorn, Walter Purkert: Felix Hausdorff biography. (Volume IB of the edition), Springer, Heidelberg 2018.
• Joachim Buhrow: A great mathematician, driven to his death by the Nazi regime in 1942. In: Wolfgang Wilhelmus : The fascist pogrom from 9./10. November 1938 - on the history of the Jews in Pomerania. Together with Julia Männchen. Colloquium of the History and Theology Sections of the Ernst-Moritz-Arndt-University Greifswald on November 2nd, 1988. Scientific contributions from the Ernst-Moritz-Arndt-University Greifswald, 1989.
• SD Chatterji: Felix Hausdorff as a measure theorist. Mathematical semester reports 49 (2002), pp. 129–143.
• E. Eichhorn, E.-J. Thiele: Lectures in memory of Felix Hausdorff. Heldermann Verlag, Berlin 1994, ISBN 3-88538-105-2 .
• M. Epple : Felix Hausdorff's Considered Epiricism. In: JJ Gray, J. Ferreiros (Eds.): Architecture of Modern Mathematics. Essays in History and Philosophy. Oxford 2006.
• Hans-Joachim Girlich : Felix Hausdorff and applied mathematics. In: Herbert Beckert , Horst Schumann (Hrsg.): 100 Years of Mathematical Seminar at the Karl Marx University of Leipzig. German Science Publishers, Berlin 1981.
• P. Koepke, V. Kanovei: Descriptive set theory in Hausdorff's basics of set theory. 2001 ( math.uni-bonn.de, PDF).
• Wolfgang Krull:  Hausdorff, Felix. In: New German Biography (NDB). Volume 8, Duncker & Humblot, Berlin 1969, ISBN 3-428-00189-3 , p. 111 f. ( Digitized version ).
• GG Lorentz: The mathematical work of Felix Hausdorff. Annual report of the DMV 69 (1967), 54 (130) - 62 (138).
• Werner Stegmaier : A mathematician in the landscape of Zarathustra. Felix Hausdorff as a philosopher. Nietzsche Studies 31 (2002), 195–240.
• Walter Purkert : Continuum Problem and Well-Order - Felix Hausdorff and the Events at the 3rd International Congress of Mathematicians in Heidelberg. In: M. Folkerts, U. Hashagen, R. Seising (Hrsg.): Form, number and order. Festschrift for Ivo Schneider. Stuttgart 2004, pp. 223-241.
• Walter Purkert : The Double Life of Felix Hausdorff / Paul Mongré. Mathematical Intelligencer, 30 (2008), 4, p. 36 ff.
• Walter Purkert: Felix Hausdorff - Paul Mongré. Mathematician - Philosopher - Man of Letters . Hausdorff Center for Mathematics, Bonn 2013.
• U. Roth: The language criticism is an act. Paul Mongré's examination of F. Mauther's “Contributions to a Critique of Language”. Journal for German Linguistics. 30, 1 (2002).
• F. Vollhardt: From social history to cultural studies? The literary-essayistic writings of the mathematician Felix Hausdorff (1868–1942): Preliminary remarks with systematic intent. In: M. Huber, G. Lauer (Ed.): After social history - concepts for a literary study between historical anthropology, cultural history and media theory. Max Niemeier Verlag, Tübingen 2000, pp. 551-573.
• S. Wagon: The Banach-Tarski Paradox. Cambridge Univ. Press, Cambridge 1993.
• Hausdorff, Felix. In: Lexicon of German-Jewish Authors . Volume 10: Güde – Hein. Edited by the Bibliographia Judaica archive. Saur, Munich 2002, ISBN 3-598-22690-X , pp. 262-268.

Web links

Commons : Felix Hausdorff  - Collection of images, videos and audio files

Wikisource: Felix Hausdorff  - Sources and full texts

• Literature by and about Felix Hausdorff in the catalog of the German National Library
• Overview of Felix Hausdorff's courses at the University of Leipzig (summer semester 1896 to summer semester 1910)
• Felix Hausdorff in the professorial catalog of the University of Leipzig
• The homepage of the Hausdorff Edition
• Felix Hausdorff's estate in the ULB Bonn
• Hausdorff-Koenig estate in the ULB Bonn
• Felix Hausdorff - Paul Mongré. Biography of Prof. W. Purkert (PDF file; 508 kB)
• Spektrum .de: Felix Hausdorff (1868–1942) November 1, 2018

Individual evidence

1. ^ Archive of the University of Leipzig, PA 547.
2. ^ E. Neuenschwander: Felix Hausdorff's last years of life according to documents from the Bessel-Hagen estate. In: Brieskorn 1996, pp. 253-270.
3. ^ Bessel-Hagen estate, Bonn University Archives. Printed in Brieskorn 1996, pp. 263–264 and in facsimile pp. 265–267.
4. ^ Walter Purkert: Farewell letter Felix Hausdorffs . In: Birgit Bergmann, Moritz Epple (ed.): Jewish mathematicians in the German-speaking academic culture . Springer, Berlin / Heidelberg 2009, ISBN 978-3-540-69250-8 , Bonn, p. 90-108 , doi : 10.1007 / 978-3-540-69252-2_7 ( Wikisource ).
5. ↑ See finding aid from the Hausdorff estate.
6. ↑ Lower Saxony State and University Library in Göttingen, Manuscript Department, Hilbert branch, No. 136.
7. ↑ Detailed information can be found in the collected works, Volume II, pp. 9–12.
8. ↑ H .: Collected Works. Volume II: Fundamentals of set theory. Springer-Verlag, Berlin, Heidelberg etc. 2002. Comments by U. Felgner, pp. 598–601.
9. ↑ H .: Collected Works. Volume II: Fundamentals of set theory. Springer-Verlag, Berlin, Heidelberg etc. 2002. pp. 604-605.
10. ↑ See the essay by U. Felgner: The Hausdorff theory of quantities and their history of effects. ${\displaystyle \eta _{\alpha }}$In H .: Collected Works. Volume II: Fundamentals of set theory. Springer-Verlag, Berlin, Heidelberg etc. 2002. pp. 645-674.
11. ^ See on this and on similar sentences by Kuratowski and Zorn the commentary by U. Felgner in the collected works, Volume II, pp. 602–604.
12. ↑ A. Schoenflies: The Development of the Doctrine of Point Manifolds. Part II. Annual report of the DMV, 2nd supplementary volume, Teubner, Leipzig 1908, p. 40.
13. ↑ For the history of the effects of Hausdorff's spherical paradox, see Collected Works, Volume IV, pp. 11-18; also the article by P. Schreiber in Brieskorn 1996, pp. 135-148, and the monograph Wagon 1993.
14. ^ P. Urysohn: Mémoire sur les multiplicités Cantoriennes. (PDF; 6.2 MB), Fundamenta Math. 7 (1925), pp. 30-137; 8 (1926), pp. 225-351.
15. ↑ P. Alexandroff: Sur la puissance des ensembles mesurables B. Comptes rendus Acad. Sci. Paris 162 (1916), pp. 323-325.
16. ^ WH Young: On the teaching of the uncompleted point sets. Reports on the negotiations of the Royal Saxon. Ges. The Wiss. to Leipzig, Math.-Phys. Class 55 (1903), pp. 287-293.
17. ↑ Alexandorff, Hopf 1935, p. 20. For more information see Collected Works, Volume II. Pp. 773–787.
18. ↑ For the history of the effects of dimension and external dimensions see the articles by Bandt / Haase and Bothe / Schmeling in Brieskorn 1996, pp. 149–183 and pp. 229–252 as well as the commentary by S. D. Chatterji in the collected works, volume IV, p. 44–54 and the literature cited there.
19. ↑ For the overall complex of these works and estate studies, see Collected Works Volume IV. Pp. 105–171, 191–235, 255–267 and 339–373.
20. ↑ See the commentary by S. D. Chatterji in the Gesammelte Werken, Volume IV, pp. 182–190.
21. ↑ H. Hahn: F. Hausdorff, set theory. Monthly books for mathematics and physics 35 (1928), 56–58.
22. ^ Hausdorffstrasse in the Bonn street cadastre
23. ↑ Council meeting of May 18, 2011 (resolution no. RBV-822/11), official announcement: Leipzig Official Gazette no. 11 of June 4, 2011, in force since July 5, 2011 and August 5, 2011. Cf. Official Journal No. 16 of September 10, 2011.
24. ↑ Academy program. ( Memento from May 18, 2015 in the Internet Archive ).
25. ↑ See new publications. At: Springer.com.
26. ↑ The DNB data also provide an overview of all volumes .
27. ↑ Review by Jeremy Gray of Volumes 1a, 3, 8, 9, Bulletin AMS, Vol. 51, 2014, 169-172.

This version was added to the list of articles worth reading on May 4, 2007 .