**Aleksandrov**'s name into the Roman alphabet. The most common way, other than Aleksandrov, is to write it as

**Alexandroff.**

Pavel Sergeevich Aleksandrov's father Sergej Aleksandrovich Aleksandrov was a medical graduate from Moscow University who had decided not to follow an academic career but instead had chosen to use his skills in helping people and so he worked as a general practitioner in Yaroslavskii. Later he worked in more senior positions in a hospital in Bogorodskii, which is where he was when Pavel Sergeevich was born.

When Pavel Sergeevich was one year old his father moved to Smolensk State hospital, where he was to earn the reputation of being a very fine surgeon, and the family lived from this time in Smolensk. The city of Smolensk is on the Dnieper River 420 km west of Moscow. Pavel Sergeevich's early education was from his mother, Tsezariya Akimovna Aleksandrova, who applied all her considerable talents to bringing up and educating her children. It was from her that Aleksandrov learnt French and also German. His home was one that was always filled with music as his brothers and sisters all had great talent in that area.

The fine start which his mother gave him meant that he always excelled at the grammar school in Smolensk which he attended. His mathematics teacher Alexsander Romanovich Eiges soon realised that his pupil had a remarkable talent for the subject and ([3] and [4]):-

In 1913 Aleksandrov graduated from the grammar school being dux of the school and winning the gold medal. Certainly at this time he had already decided on a career in mathematics, but he had not set his sights as high as a university teacher, rather he was aiming to become a secondary school teacher of mathematics. Eiges was the role model who he was aspiring to match at this stage, for Eiges had done more than teach Aleksandrov mathematics, he had also influenced his tastes in literature and the arts.... at grammar school he studied celestial mechanics and mathematical analysis. But his interest was mainly directed towards fundamental problems of mathematics: the foundations of geometry and non-euclidean geometry. Eiges had a proper appreciation of his pupil and exerted a decisive influence on his choice of a career in mathematics.

Aleksandrov entered Moscow University in 1913 and immediately he was helped by Stepanov. Stepanov, who was working at Moscow University, was seven years older than Aleksandrov but his home was also in Smolensk and he often visited the Aleksandrov home there. Stepanov was an important influence on Aleksandrov at this time and suggested that Aleksandrov join Egorov's seminar even in the first year of his studies in Moscow. In Aleksandrov's second year of study he came in contact with Luzin who had just returned to Moscow. Aleksandrov wrote (see for example [3] or [4]):-

Aleksandrov proved his first important result in 1915, namely that every non-denumerable Borel set contains a perfect subset. It was not only the result which was important for set theory, but also the methods which Aleksandrov used which turned out to be one of the most useful methods in descriptive set theory. After Aleksandrov's great successes Luzin did what many a supervisor might do, he realised that he had one of the greatest mathematical talents in Aleksandrov so he thought that it was worth asking him to try to solve the biggest open problem in set theory, namely the continuum hypothesis.After Luzin's lecture I turned to him for advice on how best to continue my mathematical studies and was struck most of all by Luzin's kindness to the man addressing him - an18-year old student ... I then became a student of Luzin, during his most creative period ... To see Luzin in those years was to see a display of what is called an inspired relationship to science. I learnt not only mathematics from him, I received also a lesson in what makes a true scholar and what a university professor can and should be. Then, too, I saw that the pursuit of science and the raining of young people in it are two facets of one and the same activity - that of a scholar.

After Aleksandrov failed to solve the continuum hypothesis (which is not surprising since it can neither be proved or disproved as was shown by Cohen in the 1960s) he thought he was not capable of a mathematical career. Aleksandrov went to Novgorod-Severskii and became a theatre producer. He then went to Chernikov where, in addition to theatrical work, he lectured on Russian and foreign languages, becoming friends with poets, artists and musicians. After a short term in jail in 1919 at the time of the Russian revolution, Aleksandrov returned to Moscow in 1920. Luzin and Egorov had built up an impressive research group at the University of Moscow which the students called 'Luzitania' and they, together with Privalov and Stepanov, were very welcoming to Aleksandrov on his return.

It was not an immediate return to Moscow for Aleksandrov, however, for he spent 1920-21 back home in Smolensk where he taught at the University. During this time he worked on his research, going to Moscow about once every month to keep in touch with the mathematicians there and to prepare himself for his examinations. At around this time Aleksandrov became friendly with Urysohn, who was a member of 'Luzitania', and the friendship would soon develop into a major mathematical collaboration.

After taking his examinations in 1921, Aleksandrov was appointed as a lecturer at Moscow university and lectured on a variety of topics including functions of a real variable, topology and Galois theory. In July 1922 Aleksandrov and Urysohn went to spend the summer at Bolshev, near to Moscow, where they began to study concepts in topology. Hausdorff, building on work by Fréchet and others, had created a theory of topological and metric spaces in his famous book *Grundzüge der Mengenlehre* Ⓣ published in 1914. Aleksandrov and Urysohn now began to push the theory forward with work on countably compact spaces producing results of fundamental importance. The notion of a compact space and a locally compact space is due to them.

In the summers of 1923 and 1924 Aleksandrov and Urysohn visited Göttingen and impressed Emmy Noether, Courant and Hilbert with their results. The mathematicians in Göttingen were particularly impressed with their results on when a topological space is metrisable. In the summer of 1924 they also visited Hausdorff in Bonn and he was fascinated to hear the major new directions that the two were taking in topology. However while visiting Hausdorff in Bonn ([3] and [4]):-

Aleksandrov and Urysohn then visited Brouwer in Holland and Paris in August 1924 before having a holiday in the fishing village of Bourg de Batz in Brittany. Of course mathematicians continue to do mathematics while on holiday and they were both working hard. On the morning of 17 August Urysohn began to write a new paper but tragically he drowned while swimming in the Atlantic later that day. Aleksandrov determined that no ideas of his great friend and collaborator should be lost and he spent part of 1925 and 1926 in Holland working with Brouwer on preparing Urysohn's paper for publication.Every day Aleksandrov and Urysohn swam across the Rhine - a feat that was far from being safe and provoked Hausdorff's displeasure.

The atmosphere in Göttingen had proved very helpful to Aleksandrov, particularly after the death of Urysohn, and he went there every summer from 1925 until 1932. He became close friends with Hopf and the two held a topological seminar in Göttingen. Of course Aleksandrov also taught in Moscow University and from 1924 he organised a topology seminar there. At Göttingen, Aleksandrov also lectured and participated in Emmy Noether's seminar. In fact Aleksandrov always included Emmy Noether and Hilbert among his teachers, as well as Brouwer in Amsterdam and Luzin and Egorov in Moscow.

From 1926 Aleksandrov and Hopf were close friends working together. They spent some time in 1926 in the south of France with Neugebauer. Then Aleksandrov and Hopf spent the academic year 1927-28 at Princeton in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on *Topology* the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes. In fact before the joint work with Hopf appeared in print, Aleksandrov had begun yet another important friendship and collaboration.

In 1929 Aleksandrov's friendship with Kolmogorov began and they ([3] and [4]):-

The year 1929 marks not only the beginning of the friendship with Kolmogorov but also the appointment of Aleksandrov as Professor of Mathematics at Moscow University. In 1935 Aleksandrov went to Yalta with Kolmogorov, then finished the work on his... journeyed a lot along the Volga, the Dnieper, and other rivers, and in the Caucuses, the Crimea, and the south of France.

*Topology*book in the nearby Crimea and the book was published in that year. The 'Komarovski' period also began in that year ([3] and [4]):-

In 1938-1939 a number of leading mathematicians from the Moscow University, among them Aleksandrov, joined the Steklov Mathematical Institute of the USSR Academy of Sciences but at the same time they kept their positions at the University.Over the last forty years, many of the events in the history of mathematics in the University of Moscow have been linked with Komarovka, a small village outside Moscow. Here is the house owned since1935by Aleksandrov and Kolmogorov. Many famous foreign mathematicians also visited Komarovka - Hadamard, Fréchet, Banach, Hopf, Kuratowski, and others.

Aleksandrov wrote about 300 scientific works in his long career. As early as 1924 he introduced the concept of a locally finite covering which he used as a basis for his criteria for the metrisability of topological spaces. He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929. His methods allowed arguments of combinatorial and algebraic topology to be applied to point set topology and brought together these areas. Aleksandrov's work on homology moved forward with his homological theory of dimension around 1928-30

Aleksandrov was the first to use the phrase 'kernel of a homomorphism' and around 1940-41 he discovered the ingredients of an exact sequence. He worked on the theory of continuous mappings of topological spaces. In 1954 he organised a seminar on this last topic aimed at first year students at Moscow University and in this he showed one of the aspects of his career which was of major importance to him, namely the education of students. This is described in ([3] and [4]):-

Many honours were given to Aleksandrov for his outstanding contribution to mathematics. He was president of the Moscow Mathematical Society from 1932 to 64, vice president of the International Congress of Mathematicians from 1958 to 62, a corresponding member of the USSR Academy of Sciences from 1929 and a full member from 1953. Many other societies elected Aleksandrov to membership including the Göttingen Academy of Sciences, the Austrian Academy of Sciences, the Leopoldina Academy in Halle, the Polish Academy of Sciences, the National Academy of Sciences of the United States, the London Mathematical Society, the American Philosophical Society, and the Dutch Mathematical Society.To the training of these students and those who came after them, Aleksandrov literally devoted all his strength. His influence on the class of young men studying topology under him was never purely mathematical, however real and significant that was. There were physical days exercise on topological walks, in long outings lasting several days by boat, ... in swimming across the Volga or other broad stretches of water, in skiing excursions lasting for hours on the slopes outside Moscow, slopes to which Aleksandrov gave striking, fantastic names...

He edited several mathematical journals, in particular the famous Soviet Journal *Uspekhi Matematicheskikh Nauk*, and he received many Soviet awards, including the Stalin Prize in 1943 and five Orders of Lenin.

Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology. After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school. On 28 December 1982 the rector issued a circular creating the Department of general topology and Geometry. Vitaly Vitalievich Fedorchuk was elected Head of the Department.

Also in memory of Aleksandrov's contributions to topology at Moscow University and his work with the Moscow Mathematical Society, there is an annual topological symposium *Aleksandrov Proceedings* held every May.

**Article by:** *J J O'Connor* and *E F Robertson*

**Click on this link to see a list of the Glossary entries for this page**