Felix Hausdorff: Difference between revisions

{{short description|German mathematician}}

| name = Felix Hausdorff

| image = Hausdorff 1913-1921.jpg

| caption =

| birth_date = {{Birth date|1868|11|08|mf=y}}

| birth_place = [[Breslau]], [[Kingdom of Prussia]]<br>(now [[Wrocław]], [[Poland]])

| death_date = {{death date and age|1942|01|26|1868|11|08|mf=y}}

| death_place = [[Bonn]], [[Nazi Germany|Germany]]

| nationality =  [[Germans|German]]

| field = [[Mathematics]]

| alma_mater = [[University of Leipzig]]

| thesis_title = Zur Theorie der astronomischen Strahlenbrechung

| thesis_year = 1891

| doctoral_advisor = {{plainlist|

| known_for = {{plainlist|

| religion =

| spouse = Charlotte Hausdorff (1873-1942)

| footnotes=

'''Felix Hausdorff''' ({{IPAc-en|ˈ|h|aʊ|s|d|ɔːr|f}} {{Respell|HOWS|dorf}}, {{IPAc-en|ˈ|h|aʊ|z|d|ɔːr|f}} {{Respell|HOWZ|dorf}};<ref>{{cite web |url = https://www.dictionary.com/browse/hausdorff-space |title = Hausdorff space Definition & Meaning |access-date = 15 June 2022 }}</ref> November 8, 1868 – January 26, 1942<ref>{{Cite web |last=Purkert |first=Prof. Dr. Walter |title=Felix Hausdorff - Paul Mongré |url=http://hausdorff-edition.de/media/pdf/FH.pdf |access-date=November 14, 2023 |website=Hausdorff Edition }}</ref>) was a German [[mathematician]], pseudonym '''Paul Mongré''' (''à mon [https://fr.wiktionary.org/wiki/gré gré]'' (Fr.) = "according to my taste"),<ref>{{cite journal|doi=10.1007/s00283-021-10083-9 |title=Biographie. Felix Hausdorff, Gesammelte Werke. Band IB. By Egbert Brieskorn and Walter Purkert |date=2021 |last1=Schubring |first1=Gert |journal=The Mathematical Intelligencer |volume=43 |issue=4 |pages=94–98 |doi-access=free }}</ref> who is considered to be one of the founders of modern [[topology]] and who contributed significantly to [[set theory]], [[descriptive set theory]], [[measure theory]], and [[functional analysis]].

Hausdorff was Jewish, and life became difficult for him and his family after the ''[[Kristallnacht]]'' of 1938. The next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Hausdorff, along with his wife and his sister-in-law, died by suicide by taking an overdose of [[veronal]], rather than comply with German orders to move to the Endenich camp, and there suffer the likely implications, about which he held no illusions.

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.

This quote emphasizes the undisguised [[anti-Semitism|antisemitism]] present, which especially took a sharp upturn throughout the German Reich after the [[Gründerkrach|stock market crash of 1873]]. Leipzig was a focus of antisemitic sentiment, especially among the student body, which may well be the reason that Hausdorff did not feel at ease in Leipzig. Another contributing factor may also have been the stresses due to the hierarchical posturing of the Leipzig professors.

After his Habilitation, Hausdorff wrote other works on [[optics]], on [[non-Euclidean geometry]], and on [[hypercomplex 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 set]]s. Initially, it was only out of philosophical interest that Hausdorff began to study [[Georg Cantor]]'s work, beginning around 1897, but already in 1901 Hausdorff began lecturing on set theory. His was one of the first ever lectures on set theory; only [[Ernst Zermelo]]'s lectures in Göttingen College during the winter of 1900/1901 were earlier. That same year, he published his first paper on order types in which he examined a generalization of [[well-ordering]]s called [[graded order types]], where a [[linear order]] is graded if no two of its segments share the same [[order type]]. He generalized the [[Cantor–Bernstein theorem]], which said the collection of countable order types has the [[cardinality of the continuum]] and showed that the collection of all graded types of an [[Idempotence|idempotent]] cardinality {{var|m}} has a cardinality of 2^{{var|m}}.<ref>{{Cite book|title = Handbook of the History of Logic: Sets and extensions in the twentieth century|url = https://books.google.com/books?id=ZF_QckMFy-oC&q=graded%2520order%2520type&pg=PA159|publisher = Elsevier|date = 2012-01-01|isbn = 9780444516213|language = en|first = Dov M.|last = Gabbay}}</ref>

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 [[Andrey Kolmogorov|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

After the takeover by the [[National Socialist]] party, [[anti-Semitism|antisemitism]] became state doctrine. Hausdorff was not initially concerned by the "[[Law for the Restoration of the Professional Civil Service]]", adopted in 1933, because he had been a German public servant since before 1914. However, he was not completely spared, as one of his lectures was interrupted by National Socialist student officials. In the winter semester of 1934/1935, there was a working session of the National Socialist German Student Union (NSDStB) at the University of Bonn, which chose "Race and Ethnicity" as their theme for the semester. Hausdorff cancelled his 1934/1935 winter semester Calculus III course on 20 November, and it is assumed that the choice of theme was related to the cancellation of Hausdorff's class, since in his long career as a university lecturer he had always taught his courses through to their end.

Hausdorff's entrance into a thorough study of ordered sets was prompted in part by Cantor's continuum problem: where should the [[cardinal number]] <math>\aleph = 2^{\aleph_0}</math> be placed in the sequence <math>\{\aleph_{\alpha}\}</math>? In a letter to Hilbert on 29 September 1904, he speaks of this problem, "it has plagued me almost like [[monomania]]".<ref>Niedersächsische Staats- und Universitätsbibliothek zu Göttingen, Handschriftenabteilung, NL Hilbert, Nr. 136.</ref> Hausdorff saw a new strategy to attack the problem in the set <math> \mathrm{card} (T(\aleph_0)) = \aleph</math>. Cantor had suspected <math>\aleph = \aleph_1</math>, but had only been able to show that <math>\aleph \geq \aleph_1</math>. While <math>\aleph_1</math> is the "number" of possible [[well-ordering]]s of a [[countable set]], <math>\aleph</math> had now emerged as the "number" of all possible orders of such an amount. It was natural, therefore, to study systems that are more specific than orders, but more general than well-orderings. Hausdorff did just that in his first volume of 1901, with the publication of theoretical studies of "graded sets". However, we know from the results of [[Kurt Gödel]] and [[Paul Cohen (mathematician)|Paul Cohen]] that this strategy to solve the continuum problem is just as ineffectual as Cantor's strategy, which was aimed at generalizing the [[Cantor&ndash;Bendixson theorem|Cantor–Bendixson principle]] from [[closed set]]s to general uncountable sets.

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.<ref>Siehe dazu und zu ähnlichen Sätzen von [[Kazimierz Kuratowski|Kuratowski]] und [[Max August Zorn|Zorn]] den Kommentar von U.&nbsp;Felgner in den gesammelten Werken, Band&nbsp;II, S.&nbsp;602–604.</ref>

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 [[Hausdorff method]]s. In [[Godfrey Harold Hardy|Hardy]]'s classic ''Divergent Series'', an entire chapter is devoted to the Hausdorff method. The classical methods of [[Otto Hölder|Hölder]] and [[Ernesto Cesàro|Cesàro]] proved to be special cases of the Hausdorff method. Every Hausdorff method is given by a moment sequence; in this context Hausdorff gave an elegant solution of the moment problem for a finite interval, bypassing the theory of continued fractions. In his paper ''Moment problems for a finite interval'' of 1923 he treated more special moment problems, such as those with certain restrictions for generating density <math>\varphi(x)</math>, for instance <math>\varphi(x) \in L^p[0,1]</math>. Criteria for solvability and decidability of moment problems occupied Hausdorff for many years, as hundreds of pages of handwritten notes in his [[Nachlass]] attest.<ref>''Gesammelte Werke Band&nbsp;IV'', S.&nbsp;105–171, 191–235, 255–267 and 339–373.</ref>

In 1938, Hausdorff's last work ''Extension of a continuous map'' showed that a [[continuous function]] from a closed subset <math>F</math> of a metric space <math>E</math> can be extended to all of <math>E</math> (although the image may need to be extended). As a special case, every [[homeomorphism]] from <math>F</math> can be extended to a homeomorphism from <math>E</math>. 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 <math>E</math> be a metric space, <math>F \subseteq E</math> a closed subset. If <math>F</math> 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 [[Kuratowski closure axioms]] up to the axiom of idempotence. These spaces are often also called closure spaces, and Hausdorff used them to study relationships between the [[Maurice René Fréchet|Fréchet]] limit spaces and [[topological spaces]].

The "Hausdorff-Edition", edited by [[Egbert Brieskorn|E. Brieskorn]]&nbsp;(†), [[Friedrich Hirzebruch|F. Hirzebruch]]&nbsp;(†), [[Walter Purkert|W. Purkert]] (all Bonn), [[Reinhold Remmert|R. Remmert]]&nbsp;(†) (Münster) and [[Erhard Scholz|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 Science+Business Media|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 [https://web.archive.org/web/20051119053418/http://www.aic.uni-wuppertal.de/fb7/hausdorff/ website of the Hausdorff Edition (German)] for further information. The volumes are: