# Bronisław Knaster

### Quick Info

Warsaw, Russian Empire (now Poland)

Wrocław, Poland

**Bronisław Knaster**was a Polish mathematician who was known for his work in point-set topology.

### Biography

**Bronisław Knaster**was born in Warsaw but at the time of his birth, Warsaw was not a major city of Poland since Poland had officially ceased to exist in 1795 when its lands were divided between Prussia, the Austrian Empire and the Russian Empire. In 1893 Warsaw was in the Vistula Province of the Russian Empire but, with Russia trying to eradicate Polish culture and language, it saw much unrest as Poles tried to assert their identity. Bronisław's father, Ludwik Knaster, was a much respected hospital doctor in Warsaw, the head of the therapeutic clinic at the 'Baby Jesus' hospital and president of the Association of Physicians.

School education proved difficult for young Knaster who entered the first class of the Fourth Junior High School in 1902. Beginning in autumn of 1904 there were school strikes as pupils and their parents protested against school teaching being in Russian and pupils being even forbidden to speak Polish in school. When school strikes occurred, the Russian authorities reacted by closing the school and the Fourth Junior High School was closed. It appears that the Russian authorities were happy to close schools and keep illiteracy in Poland as high as possible. After a year being out of formal education, in January 1906 he was admitted to the School of the Teachers' Association, and after it was closed by the authorities in 1907, he was admitted as a student in the fifth grade of the Jan Kreczmar School. After the closure of this school in May 1908 and the arrest of many of its students, Knaster moved to the Second Boys Junior High School, at which he completed his school education in 1911 with a silver medal. On his Russian matriculation certificate there is no Polish language, but he had passes in the following subjects (with the grade out of 5): religion (5), Russian language and literature (5), philosophy (5), Latin (4), mathematics (5), physics (5), mathematical geography (5), geography (4), German (5), French (5), jurisprudence (5). Despite the difficulties with his education we see that he graduated in 1911 with excellent grades and looked for a university education.

The University of Warsaw, his home town university, should have been a natural choice but, like the schools, its teaching was in Russian and the staff were entirely Russian, so patriotic Poles boycotted it. As a consequence most Poles from Vistula Province at this time went abroad for their university education. Some went to Galicia where, although under Austrian control, Polish education still flourished. Knaster's father, however, had studied in Paris so it was natural for his son to follow suit. One might imagine that Paris was an ideal city for someone to obtain an outstanding mathematical education but at this stage Knaster was looking to follow his father and obtain a medical degree.

In his first year in Paris, 1911-1912, before beginning his medical degree he studied physics, chemistry and natural science, passing the examinations in these topics and thus qualifying to enter the Faculty of Medicine. After two years studying medicine, sat and passed the examinations in March 1914. During his studies in Paris, he met Maria Morska, who later became a well-known author and respected journalist writing under the pseudonym Mariusz Dawn. In July 1914 they returned to Poland (still of course a part of the Russian Empire) for a holiday and intending to get married. By the end of July 1914, however, France had begun mobilizing its troops and, on 3 August, Germany declared war on France. Returning to France was not possible but the three years he had spent in Paris were important ones for his education and he had become familiar with the French language and culture, both of which he loved. Knaster married Maria Morska in Dabrowa near Tarnów in October 1914.

Knaster took a job in the Judicial Department of the Commissariat of the Fourth Civil Guard District of the City of Warsaw. The German advance into Poland during World War I saw the Russians leave Warsaw in August 1915 and a German governor was installed. Germany, seeking to bring Poles onside, allowed a certain amount of restoration of Polish education and the Poles were allowed to organise a new Polish University of Warsaw. They did this in a remarkably short period of time and the university opened in November 1915. Teaching was in Polish but, in order to keep control of the patriotic movement, the Germans only allowed 50 professors although the number of students were unlimited. Knaster enrolled and began studying logic under Jan Lukasiewicz who had been invited to the new University of Warsaw when it reopened. Quite quickly, however, he changed from logic to studying mathematics. Frequently asked later why he did not continue his medical studies, he liked to say that he made the change because of the purity of mathematical methods and the certainty of its results, which in his eyes was very different from medicine.

Mathematics in the new University of Poland was strong from the beginning. Zygmunt Janiszewski led the new centre along with Stefan Mazurkiewicz. Wacław Sierpiński joined then in 1918. Kazimierz Kuratowski writes in [1]:-

As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field. The meeting of that seminar, taken up to a large extent with sometimes quite vehement discussions between Janiszewski and Mazurkiewicz, were a real intellectual treat for the participants.To understand the remarkable mathematical environment that Knaster had entered we quote again from Kazimierz Kuratowski [1]:-

In the first volume of [Polish Science: Its Needs, Organisation, and Development], which appeared in 1918, Janiszewski published an article "On the needs of mathematics in Poland", which with amazing clarity and precision presented a blueprint for Polish mathematics. Janiszewski started out with the assumption that Polish mathematicians do not have to be satisfied with the role of followers and customers of foreign mathematical centres but can achieve an independent position for Polish mathematics. One of the best ways of achieving this goal, suggested Janiszewski, was for groups of mathematicians to concentrate on relatively narrow fields in which Polish mathematicians had common interests and - more importantly - had already made internationally important contributions. These areas included set theory, and the foundations of mathematics.The first students to enter into this high powered mathematical environment were Knaster, Kazimierz Kuratowski and Stanisław Saks. Knaster's studies were interrupted, however, by the outbreak of the Polish-Soviet war of 1920. An independent Poland had been declared on 11 November 1918 and the Treaty of Versailles of 1919 defined much of the boundaries of the new Poland but left the problem of the eastern boundary vague. Poland looked to regain its lands in the east while the Soviets looked to expand to the west and regain lands they held prior to World War I. Border incidents took place during 1919 but in March 1920 a full scale war broke out when Polish troops advanced towards Kiev. A Soviet counter offensive saw their armies advance towards Warsaw in July 1920 and Knaster volunteered to service in the Polish army. He served as a corporal stretcher-bearer until November 1920 and was later decorated with the Legion Cross by Marshall Piludski. He paid a heavy price, however, for he contracted a rare form of malaria which caused him almost unbearable physical problems for ten years.

Back at the University of Warsaw, Knaster continued to undertake research advised by Stefan Mazurkiewicz and soon published his first papers. In 1921, jointly with Kazimierz Kuratowski, he published the paper

*Sur les ensembles connexes*Ⓣ and, in the following year, his paper, joint with Waclaw Sierpinski,

*Sur une ensemble abstrait, dont chaque élément est un élément limite de chaque sous ensemble non dénombrable*Ⓣ, was published. These papers were, respectively, in the second and the third volumes of the new Polish journal

*Fundamenta Mathematicae*. Knaster was interested in continua, defined technically as compact connected metric sets. Janusz Charatonik writes [5]:-

The first curiosities in the structure of continua were observed in 1910 by L E J Brouwer who constructed the first example of an indecomposable continuum, i.e. a continuum which is not the union of two of its proper subcontinua. Two years later, Z Janiszewski announced that there exists a curve containing no arc. Both these examples inspired Knaster and Kuratowski to ask the question: Does there exist an indecomposable continuum whose every subcontinuum is also indecomposable? Obviously, any such a continuum, if it existed, could not contain an arc, so it had to differ substantially from any continuum known so far. Using very hard, complicated, and purely geometric methods, Knaster succeeded in constructing such an example. This was his Ph.D. thesis 'Un continu dont tout sou-continu est indécomposable' Ⓣ (1922).Knaster received his doctorate in 1923, becoming one of the first doctors of the new Polish University of Warsaw. With such outstanding results Knaster's future career looked set for the highest level. Instead of years of triumph, however, the next few years became more and more difficult due to the malaria he had contracted during the Polish-Soviet war. Quinine taken orally proved to be completely ineffective and the attacks of malaria began to come more and more often, and lasted longer and longer. Under these conditions, it was decided that he would feel better in the south, especially since a few months stay in a special sanatorium near Dresden did not help. In 1924 he left for Italy, where he felt well enough to work. From then on, for the next few years, with short breaks, he lived, hoped to recover and worked there. The malaria, however, did not leave him while he lived in Italy, and his subsequent attacks gradually became so dangerous that over time they began to require that he received treatment in hospital. A pure piece of luck saved him: one day he picked up a newspaper which contained an article about the treatment of some varieties of malaria with intramuscular injections of quinine. Within a year around 1930 the disease passed, leaving him, however, with a severely damaged body and he struggled with his health for the rest of life and was sometimes forced to take long stays away from home.

The problems with his health that we have just recounted might lead one to expect that his publication record in the nine years from 1921 to 1930 would be much poorer that that from 1930 to 1939. In fact a look at his papers shows that in fact the situation is quite the opposite with about three times as many papers in the first period than in the second. The answer lies in the fact that, because of his health problems, he was unable to teach and take on the many other duties that a university professor has to undertake. When his health was at its worst, all he was able to do was undertake research.

He habilitated in 1926, giving him the title of associate professor. He attended the International Congress of Mathematicians held in Bologna in September 1928 and gave two lectures

*Sui punti regolari nelle curve di Jordan*Ⓣ and

*Decomposizioni continue e semicontinue nell' Analysis situs*Ⓣ. While at this Congress, the Polish mathematicians arranged for Ernst Zermelo to make a lecture tour of Poland in 1929. Knaster offered to have Zermelo stay at his apartment while in Warsaw and Zermelo accepted. In fact Knaster organised everything concerning the Warsaw visit. In his 1930 report of his visit, Zermelo says he greatly appreciated the hospitality and discussions with Knaster. Knaster clearly greatly enjoyed Zermelo's visit and the two corresponded between August 1929 and October 1932. Knaster wrote:-

I have grown so fond of your scientific points of view and of your further remarks and aphorisms that I miss you ...With his health improving, Knaster had returned to Warsaw in early 1929 and began to run his advanced seminar in topology, taught courses, took on editorial duties and other organizational activities. He became an editor of the new series of

*Mathematical Monographs*, which was founded in 1931. He attended the International Congress of Mathematicians held in Zurich in September 1932 and gave the lecture

*Über unikohärente Kontinua*Ⓣ. Also at this Congress he delivered his colleague Karol Borsuk's lecture

*Über die Zerlegung einer euklidischen n-dimensionalen Vollkugel in n Mengen*Ⓣ. We do not know why he delivered Borsuk's lecture, since Borsuk was certainly at the Congress.

We quote from Roman Duda's biography [7]:-

Friends recall that Knaster's favourite habit in the thirties was long evening walks, during which he liked to lead passionate disputes about mathematics but not only about mathematics. Outstanding musicality and great music culture, extensive literary reading, knowledge of the world resulting from his many years abroad, and above all a lively mind and sharp tongue made contacts with him refreshing and stimulating. Through his wife Maria he established close relations with the Skamander poetic group, and thanks to innate wit and intelligence and love of a lively and apt word, he won the admiration and friendship of Tuwim, Wittlin, Slonimski, eminent Polish poets and writers.On 1 September 1939 Germany invaded Poland and their forces rapidly advanced towards Warsaw. Knaster and his wife, and many of his fellow Poles, fled to the east to Lwów (now Lviv in Ukraine). On 22 September, Russian troops entered Poland from the east, five days later Warsaw fell to the Germans and, following the Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union, Poland was partitioned between these two powers. The Polish University of Lwów, the Jan Kazimierz University, was at first allowed to continue as before by the Soviets but on 18 October 1939 the Polish head of the university was replaced by a Ukrainian. On 8 January 1940 the Jan Kazimierz University was renamed the Ivan Franko Lviv State University and many of the Polish staff were replaced by Ukrainians and Russians. Knaster, however, was allowed to teach there and became a professor in the Chair of Geometry, headed by Stanislaw Mazur.

On 22 June 1941 Germany broke the non-aggression pact and invaded the Soviet Union. By July 1941 they had captured Lwów and they closed the university. Many Polish staff of the university and their families were massacred by the Germans but Knaster and his wife survived. The German occupation of Poland resulted in Poles being packed into small spaces and as a consequence there was a typhus epidemic. Lice carrying typhus filled these confined spaces. A vaccine was produced in Rudolf Weigel's Institute in Lwów and this was now in great demand. Knaster was given the job of feeding lice at Rudolf Weigel's Institute in Lwów. He had the [6]:-

... rather unusual job of spending several hours a day in the institute with a box of lice attached to his forearm [which] in those gloomy days was a rather good one.In fact it meant that he was given an Identity Card stamped with "Institute of Typhus" which provided relative security for him. In 1944 the Soviet forces again took Lwów and the Ukrainian University opened again. Knaster was offered his position as a professor in the Chair of Geometry but, after the new Polish borders were established following the end of World War II in 1945, he decided to leave Lwów, which was no longer in Poland, and move to Cracow with its famous Jagiellonian University. He lectured there and assisted in restarting printing so that volume 33 of

*Fundamenta Mathematicae*was published in December 1945, an important symbol in the revival of Polish mathematics.

Not only did the Polish universities in Cracow, Warsaw and Poznan reopen in late 1945 but four new Polish universities in Wrocław, Lublin, Lódz and Torum opened. Knaster was now head hunted by all four of these new universities and offered professorships. This presented him with a hard choice but, after some deliberation, he decided to accept Wrocław [6]:-

When Knaster came to Wrocław, he left behind 52 years full of events and dramas, including a cruel war which brought the painful death of his mother and tragic death of his beloved wife. A mathematician of world renown, an energetic organiser of scientific life, an experienced mathematical editor - Knaster was a man of greatest importance for the new university. He married there Regina Szpalerska, the widow of an underground Home Army soldier, and soon devoted himself completely to didactical, organising and editorial tasks. He was not alone in that. Beside him there were also E Marczewski, H Steinhaus, and W Slebodzinski. These four men, "the great four", were pioneers of the Polish mathematical life in Wrocław.One of Knaster's passions was editorial work. He was one of the founders of the Wrocław Scientific Society in 1946 and was a board member for many years. The Society published Series B and between 1949 and 1978 he edited 131 volumes in that series. He continued to publish high quality papers, especially in the 1950s. Duda writes [6]:-

... he was a deeply good man. He loved mathematics and university life above all and it sufficed, either for a student or college official, to reveal a similar passion to acquire his most vivid sympathy. Intolerable to a passive attitude of mind, he strongly favoured creative work of a very concrete type. Everywhere - in his garden, where he tried to get new sorts of roses, in editorial work, where he exhibited perfectionism difficult to meet, in music, which he liked to play for himself, in conversation, where he admired a sharp wit and verbal dispute, and above all in mathematics, his greatest love. Favouring creativity, he disliked history, philosophy and all kinds of reflection upon the past and dead. With all his sensitivity to a talent, he highly treasured a character in a man. He admired Bertrand Russell and Albert Einstein ...At age 85 he suffered a cerebral haemorrhage and although he made a partial recovery, his last eighteen months were ones in which he suffered greatly, only speaking and moving with difficulty. He died at home in Wrocław.

We have said little of Knaster's outstanding mathematical contributions after the work for his thesis. We refer to [5] and [6] for detailed information on this. We must, however, mention his procedure of fair division which he discovered in 1945 and published in

*Sur le problème du partage pragmatique de H Steinhaus*Ⓣ (1947) and in a joint paper with Hugo Steinhaus

*Sur le partage pragmatique*Ⓣ (1947). For some details of this division problem, see THIS LINK.

### References (show)

- C E Aull and R Lowen (eds.),
*Handbook of the History of General Topology*(Springer Science & Business Media, 1997). - C Borgers,
*Mathematics of Social Choice: Voting, Compensation, and Division*(SIAM, 2010). - H-D Ebbinghaus,
*Ernst Zermelo: An Approach to His Life and Work*(Springer Science & Business Media, 2007). - K Kuratowski,
*Half a century of Polish mathematics*(Warsaw, 1973).**Papers:** - J J Charatonik, The works of BronisBaw Knaster (1893-1980) in Continuum Theory,
*C E Aull and R Lowen (eds.), Handbook of the History of General Topology*, Volume 1 (Kluwer Academic Publishers, Dordrecht, 1997), 63-78. - R Duda, Life and work of Bronislaw Knaster (1893-1980),
*Colloquium Mathematicum***51**(1987), 85-102. - R Duda, Bronislaw Knaster (1893-1980) (Polish), in
*Yearbook of the Polish Mathematical Society***XXV**(1983). - M A Jones, Connecting Fair Division and Game Theory through the Optimization of Knaster's Procedure,
*Problems, Resources, and Issues in Mathematics, Undergraduate Studies***13**(4) (2003), 321-336. - B Knaster, Zygmunt Janiszewski (on the 40th anniversary of his death) (Polish),
*Wiadomosci matematyczne*(2) 4 (1960), 1-9.

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Written by J J O'Connor and E F Robertson

Last Update April 2020

Last Update April 2020