**Karol Borsuk**'s father, Marian Borsuk, was a surgeon in Warsaw while his mother was Zofia Maciejewska. Karol attended school in Warsaw but with World War I beginning before Karol had begun secondary school, major changes occurred in Poland.

In August 1915 the Russian forces which had held Poland for many years withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly a strong school of mathematics grew up in the University of Warsaw, with topology being one of the main topics. This was begun by Janiszewski and Mazurkiewicz then carried further by Kuratowski and Sierpinski. When Borsuk entered the University of Warsaw to study mathematics it was an exciting centre for research in topology; Saks and Mazurkiewicz were both teaching at the University and making major advances in the topic.

Borsuk obtained his master's degree from the University of Warsaw in 1927 and then, from 1929, he taught at the University. He continued to study for his doctorate, under the supervision of Mazurkiewicz, and this was awarded in 1930 for his dissertation *Sur les rétractes* (published in 1931) in which he invented the theory of retracts. On 26 April 1936 Borsuk married Zofia Paczkowska. They would have two children, both girls.

The second centre for mathematical research in Poland at this time was Lvov. Borsuk visited there and during one such visit began a collaboration with Ulam. In [3] Ulam describes their joint mathematical projects:-

We started collaborating from the first. From him I learned about the truly geometric, more visual, almost "palpable" tricks and methods of topology. Our results were published in a number of papers which we sent to Polish journals and to some journals abroad. Actually my first publication in the United States appeared while I was in Lvov. It was a joint paper with Borsuk, published in the Bulletin of the American Mathematical Society. We defined the idea of "epsilon homeomorphism" - approximate homeomorphisms - and the behaviour of some topological invariants under such more general transformations - continuous ones, but not necessarily one to one. A joint paper on symmetric products introduced an idea that modifies the definition of Cartesian product and leads to the construction of some curious manifolds.

In Lvov Borsuk joined the mathematicians in the Scottish Café and contributed to the open problems which they wrote down in the famous book. In fact in 1938 he posed questions about the Hilbert cube and doing so, as Keesling remarks in [1]:-

Borsuk showed exceptional insight in conceiving of this result so early in his career.

You can see a picture of the Scottish Café.

Borsuk's career was of course about to be interrupted by the start of World War II. After the German invasion of Poland in 1939, life there became extremely difficult. There was a strategy by the Germans to put an end to the intellectual life of Poland and in attempting to achieve this aim they sent many academics to concentration camps and murdered others. The Poles had experience of surviving such attacks, however, and they employed the same tactics as they had during the period of Russian domination and they organised an underground university in Warsaw. Kuratowski writes in [2]:-

Almost all our professors of mathematics lectured at these clandestine universities, and quite a few of the students then are now professors or docents themselves. Due to that underground organisation, and in spite of extremely difficult conditions, scientific work and teaching continued, though on a considerably smaller scale of course. The importance of clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses.

During the Nazi occupation Borsuk tried to keep the University of Warsaw functioning in the way that Kuratowski describes in the quote above. However, he was unable to do this without being found out by the Nazis and he was imprisoned for this "crime". Borsuk, however, escaped and was able to survive by remaining in hiding throughout the rest of the war.

By the end of World War II the whole educational system had been destroyed and so needed to be completely rebuilt. Borsuk and Kuratowski both played important roles in this reconstruction. Borsuk continued in his post in the University of Warsaw, a post he held throughout his career, being promoted to professor in 1946. He spent some time in the United States with a number of one year appointments. He was at the Institute for Advanced Study at Princeton during session 1946-47, the University of California at Berkeley during session 1959-60, and the University of Wisconsin at Madison during session 1963-64.

In Warsaw, Borsuk led a seminar in which he developed a unique atmosphere of successful international cooperation. He influenced strongly the development of the whole area of infinite-dimensional topology with his theory of retracts and his theory of shape which became major topics for discussion at his seminar.

Borsuk introduced the important concept of absolute neighbourhood retracts in his doctoral dissertation, published in 1931, which was to lead to new and fruitful ideas in metric differential geometry, see [11] for details. In 1936 he introduced the notion of cohomotopy groups, see [8] for details, which could be said to mark the beginning of stable homotopy theory. Also in [8] Borsuk's concept of the divisor of a map, which he introduced in 1956, is discussed. The paper [12] describes Borsuk's work in shape theory from the time he introduced the concept in 1968. The notion had immediate impact on other topologists, and in particular on Borsuk's own students. Shape theory grew up at the same time as infinite-dimensional topology and the interaction between the two fields was of great mutual benefit. The author of [11] emphasises the importance of the many deep questions which Borsuk posed which stimulated most of the top mathematicians working in the area.

Many honours were given to Borsuk for his remarkable contributions. He became vice director of the Mathematical Institute of the Polish Academy of Sciences in Warsaw in 1956. He was granted honorary life membership in the Polish Mathematical Society in 1978 and the address, relating the importance of his mathematical contributions, given on that occasion is reproduced in [13]. The year 1978 is also that in which Borsuk organised the International Conference on Geometric Topology, held in Warsaw, which [1]:-

... demonstrated his widespread and profound influence on topology and the high regard in which he was held.

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

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