# Mikhail Yakovlevich Suslin

### Quick Info

Born
15 November 1894
Krasavka, Balashov, Saratov Oblast, Russia
Died
21 December 1919
Krasavka, Balashov, Saratov Oblast, Russia

Summary
Mikhail Suslin was a Russian mathematician who worked in general topology and descriptive set theory.

### Biography

Mikhail Yakovlevich Suslin's parents were Yakov Gavrilovich Suslin (born 1858) and Matrena Vasil'evna (born 1859). They were peasant landowners living in a small house next to which was a shed from which they sold small goods such as salt, paraffin, matches, spoons, buttons, thread, and horse harnesses. Mikhail Yakovlevich, known to friends and family as Mishanya, was his parents' only child. He helped his father keep the books for the shop, kept the garden free of weeds, dug the potato crop in the autumn, and went with his parents to the local Orthodox church.

He entered Krasavka School No 2 in the autumn of 1903 at the age of nine. At this school, situated across the road from the Suslins' home, he was taught by Vera Andreevna Teplogorskaya-Smirnova who quickly recognised Suslin's potential. She deserves great credit, not only for spotting his talents in the few months he studied there, but also in persuading the better-off people in the village to provide funds to allow Suslin to continue his education. He graduated from the Krasavka primary school in the spring of 1904 and entered the preparatory class of the private boy's high school in Balashov in the following year. This private school only opened in 1905 and, five years later while Suslin was still a pupil, it became a state grammar school.

While attending the school in Balashov, Suslin lodged at the home of a merchant by the name of Bezborodov. He supported himself financially by acting as a private tutor to the children of some of the wealthy citizens of Balashov. He had plenty of time to do this since he found his school work rather simple. He graduated from the high school on 30 May 1913 and was presented with the gold medal. His report reads (see [5]):-
In view of his constant excellent behaviour and diligence and his excellent successes in sciences, especially in mathematics, the Pedagogical Council have decided to award him a gold medal ...
In the summer of 1913, Suslin applied to the Department of Mathematics in the Faculty of Physics and Mathematics of Moscow University and his application was approved. In his first year of study he attended student seminars led by the differential geometer Dimitri Fedorovich Egorov. Beginning in session 1914-15, Suslin began to work with Nikolai Nikolaevich Luzin who had just returned to Moscow after studying at Göttingen for several years under Edmund Landau. Suslin was not the only student joining Luzin's group, for D E Menshov, A Ya Khinchin and P S Aleksandrov also joined. This group of outstanding young mathematicians provided an extraordinarily creative research environment. P S Aleksandrov writes in [2] about Suslin as he was setting out on his academic career:-
Even at the beginning of his student years, Suslin proved to be an interesting and picturesque person. Already at the age of 18 or 19 he had made a special plan for his future intellectual development. Mathematics was only the beginning of this plan. Physics and chemistry were to be the second stage and were to be followed by biology. Medicine was to be the final part of this plan, and to this subject Suslin intended to devote his whole future life.
Although only in his second year as an undergraduate, Suslin began to undertake research. Luzin suggested that his students work on Borel sets and asked Suslin to read Henri Lebesgue's 1905 paper Sur les functions representables analytiquement . At the same time Suslin was working with P S Aleksandrov on a problem suggested by Luzin, namely investigating whether the converse of a result found by Aleksandrov was true. Aleksandrov had proved that every Borel set can be obtained by applying an operation (named an A-operation by Suslin) to a closed set. Aleksandrov writes in [2]:-
Suslin proposed that my new set-theoretical operation should be called the A-operation and that the sets obtained by applying it to closed sets should be called A-sets. He emphasized that he was proposing this terminology in my honour, by analogy with Borel sets, which by then were usually called B-sets.
While reading Lebesgue's paper Suslin discovered that one of the lemmas (stated by Lebesgue without proof) was false. He was able to construct a counter-example to the lemma but, after further thought, he was able to use a similar method to attack the problem of whether every $A$-set is a $B$-set. Aleksandrov writes [2]:-
I spent the whole winter of 1915-16 and the whole next summer trying to prove that [result] .... My extremely persistent speculations only ceased when it became known early in the autumn of 1916 that Suslin had constructed during that summer an example of an A-set that is not a B-set and so had inaugurated a new stage in the development of the whole descriptive theory of sets.
In 1924 (five years after Suslin's death), $A$-sets were renamed Suslin sets by Felix Hausdorff in his new edition of Grundzüge der Mengenlehre . This has now become standard terminology.

Suslin graduated from Moscow University in 1917 with the top grade in every examination he took. In March 1917 Luzin requested that Suslin carry on his studies so that he might gain a professorship. He wrote (see [5]):-
I have the honour to humbly ask the Faculty of Physics and Mathematics that Mikhail Yakovlevich Suslin, who graduated with the Diploma of the first degree, be given leave to remain at the university for two years without pay in preparation for a professorship. During his studies at the university Suslin was mainly interested in the theory of functions of a real variable. Concerning this subject he studied in detail the works of Hausdorff, Baire and Lebesgue as well as following an appropriate special course. As a result of his systematic study he made the discovery of an important class of non-measurable sets which are definable in a finite way. Their existence until then had been denied by the French mathematical school following the errors in the classical memoir of Lebesgue which were also revealed by Suslin. This work of his, which has attracted general attention, and which in my view has many interesting mathematical and philosophical consequences, was published in 'Comptes Rendus' of the Paris Academy on 8 January that year. Suslin knows the French and German languages. A reprint of his work together with an instruction for future studies is attached herewith.
The request was granted but Suslin became worried about his future, with no financial assistance for his studies. It seems that Suslin's health was also causing him concern. Luzin wrote again in September 1917 requesting a studentship to support him during the two years. However, the Russian Revolution was now causing problems for everyone working in Moscow and Luzin decided that he would be able to work better in a quieter place [2]:-
In 1918 Luzin moved for a while to Ivanovo (which was then still called Ivanovo-Voznesensk). Acting on his advice, A Ya Khinchin, D E Menshov and M Ya Suslin also moved there and, like Luzin, taught at the Ivanovo Polytechnic Institute. Suslin, however, did not get on well at Ivanovo and soon lost his job there.
We need to look more deeply at why Suslin "did not get on well at Ivanovo". The problems were basically health problems, made considerably worse by severe food shortages. Even before going to Ivanovo, Suslin explained that he was ill but was still persuaded to take up a teaching position there. His teaching was outstanding. Vladimir Semenovich Fedorov writes [3]:-
As a professor Suslin left in his audiences the clear and definite memory of his distinct and rigorously paced lecturing, being infallibly methodical, being able to make students work and having compassion to the needs and demands of the audience.
The winter of 1918-19 was difficult in Ivanovo. The severe cold and lack of food exacerbated Suslin's health problems. On 14 June 1919 he requested leave for the whole summer vacation so that he could return to his parents' home and recover his health. After being told that he could take only one month holiday and must then return to the Ivanovo Polytechnic Institute to teach summer revision classes, he tended his resignation on 20 June. In his resignation letter he again explains his health problems (see [5]):-
... my doctors have found that I have a lung condition, confirmed in writing, which demands treatment by increased nourishment, and by my moving, at least for the summer period, to one of the provinces abundant in grain crops.
On learning of his resignation thirty-five students sent to the Council of the Polytechnic Institute a letter requesting that Suslin continue to teach at the Institute (see [5]):-
We students, having learned that M Ya Suslin has asked to resign, herewith state to the Council of Professors that in the person of Mikhail Yakovlevich we have a wonderful teacher. Our studies with him have been extremely interesting and rich in content. We students have attended his classes with great joy. Those of us who have had temporarily to stop studying under him regret it very much, and are hoping to continue to study under his guidance. Therefore we all wish to have M Ya Suslin among our lecturers in the future, and we ask for the cooperation of the Council of Professors.
Of course Suslin could not resign immediately so had to teach until 1 September. Moves were made to help him but, as P S Aleksandrov records, these were unsuccessful [2]:-
... V V Golubev and I I Privalov initiated a plan to appoint Suslin to a professorship at the University of Saratov. A recommendation from Luzin was expected. But he did not give it and did not support Suslin for a teaching post at Saratov. When Suslin did not get the post, he went away to his home in the country (in the Province of Saratov). He soon caught typhus and died. This was one of the most tragic pages in the history of Soviet mathematics. Until the end of Luzin's life a portrait of Suslin stood on his desk, the only portrait of Suslin that I have seen.
Two further short papers by Suslin appeared after his death. The paper Probleme 3 (1920) posed a problem which remained unsolved for forty years. The other paper Sur un corps non denombrable de nombres reels. Redige d'apres un memoire posthume de Michel Souslin par C Kuratowski appeared in 1923.

### References (show)

1. P S Aleksandrov, Pages from an autobiography (Russian), Uspekhi Mat. Nauk 34 (6) (1979), 219-249.
2. P S Aleksandrov, Pages from an autobiography, Russian Math. Surveys 34 (6) (1979), 267-302.
3. V S Fedorov, Mikhail Yakovlevich Suslin, Izv. Ivanovo-Voznesensk. Politekhn. Inst. 4 (1921), 9-10.
4. V I Igoshin, A short biography of Mikhail Yakovlevich Suslin (Russian), Uspekhi Mat. Nauk 51 (3) (1996), 3-16.
5. V I Igoshin, A short biography of Mikhail Yakovlevich Suslin, Russian Math. Surveys 51 (3) (1996), 371-383.
6. V A Uspenskii and V G Kanovei, M Ya Suslin's contribution to set-theoretic mathematics (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (5) (1988), 22-30; 103.
7. V A Uspenskii and V G Kanovei, M Ya Suslin's contribution to set-theoretic mathematics, Moscow Univ. Math. Bull. 43 (5) (1988), 29-40.

Written by J J O'Connor and E F Robertson
Last Update March 2011