February 02, 2012
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Kazimierz Kuratowski (February 2, 1896 – June 18, 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.

 Kazimierz Kuratowski was born in Warsaw on February 2, 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewski. Kuratowski was born a subject of Tsarist Russia. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian. He completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Kuratowski restarted his university education at the Warsaw University, this time in mathematics, obtaining a Ph.D. in 1921. His thesis dissertation consisted of two parts. One was devoted to an axiomatic depiction of topology due to an introduction of the closure axioms (Sur la notion de l'ensemble fini, "Fundamenta Mathematicae", 1/1920). The second one was a final resolution of an irreducible continuum, which was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Wacław Sierpiński. In autumn 1921, Kuratowski gained his PhD on the basis of the thesis, which was a solution to certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin.

Two years later Kuratowski was appointed deputy professor of mathematics at Warsaw University. He was then appointed professor of mathematics in 1927, at Lviv Polytechnic in Lviv, Poland, where he was a head of the Mathematics Department until 1933. Kuratowski was also dean of the Department of Mathematics twice there. In 1934, after the closure of the department, he was appointed a professor at Warsaw University. A year later Kuratowski was nominated as the head of the Mathematics Department and he kept his position until 1952. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. During World War II, he gave lectures at the underground university in Warsaw. In 1929, Kuratowski became a member of the Warsaw Scientific Society, where from 1946 he was vice-president at the Mathematics Department, and from 1949 he was chosen to be the vice-president of the Warsaw Scientific Society.

In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, and in 1952 of the Polish Academy of Sciences, where he was the vice-president from 1957 to 1968. After World War II, Kuratowski was actively involved in the rebuilding of scientific life in Poland. Therefore, he helped to establish the State Mathematical Institute, which was incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kazimierz Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, and was also a long-time chairman of the Polish and International Mathematics Societies. He was president of the Scientific Council of the State Institute of Mathematics (in the years 1968 - 1980). From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski.

Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lviv's Scottish Café. He was a president of the Polish Mathematical Society (PTM) and a member of the Warsaw Scientific Society (TNW). What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin. He was also one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences (IMPAN). High quality research monographs of the representatives of Warsaw's and Lviv’s School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, for instance, the Royal Society of Edinburgh, Austria, Germany, Hungary, Italy and the Union of Soviet Socialist Republics (USSR).

Kuratowski’s research mainly focused on abstract topological and metric structures. He implemented the closure axioms (known in the world as the Kuratowski closure axioms), which was fundamental for the development of topological space theory and irreducible continuum theory between two points. The most valuable results, which were obtained by Kazimierz Kuratowski after the war are those that concern the relationship between topology and analytic functions (theory), and also research in the field of cutting Euclidean spaces. Together with Ulam, who was Kuratowski’s most talented student during the Lviv Period, he introduced the concept of so-called quasi homeomorphism that opened up a new field in topological studies. Kuratowski’s research in the field of measure theory, including research with Banach, Tarski, was continued by many students. Moreover, with Alfred Tarski and Wacław Sierpiński he provided most of the theory concerning Polish spaces (that are indeed named after these mathematicians and their legacy). Knaster's and Kuratowski's brought a comprehensive and precise study on connected components theory. It was applied to issues such as cutting plane, with the paradoxical examples of connected components.

Kuratowski is the author of theorem, so-called Kuratowski - Zorn lemma, which was proven for the first time in 1922 by Kuratowski (Fundamenta Mathematicae, vol.3), which has considerable use in many (basic) theorems. Zorn gave its application in 1935 ("Bulletin of the American Mathematical Society", 41). Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminology and symbolism. His contributions to mathematics include:

  • a characterization of Hausdorff spaces which are now called Kuratowski closure axioms;
  • proof of the Kuratowski-Zorn lemma;
  • in graph theory, the characterization of planar graphs now known as Kuratowski's theorem;
  • identification of the ordered pair (x,y) with the set {{x},{x,y}};
  • the introduction of the Tarski - Kuratowski algorithm;
  • Kuratowski's closure - complement problem;
  • Kuratowski's free set theorem;
  • Kuratowski convergence of subsets of metric spaces;
  • the Kuratowski, Ryll - Nardzewski measurable selection theorem;

Kuratowski’s post-war works were mainly focused on three strands:

  • The development of homotopy in continuous functions.
  • The construction of connected space theory in higher dimensions.
  • The uniform depiction of cutting Euclidean spaces by any of its subsets, based on the properties of continuous transformations of these sets.

Among over 170 published works are valuable monographs and books including Topologies (Vol. I, 1933, Vol II, 1950), the important work, published in English and Russian on set theory (together with Andrzej Mostowski, Edition I, 1952), an introduction to set theory and topology. He was a founder of studies in mathematical analysis titled “Half a century of Polish mathematics 1920 - 1970” (1973) and "Notes to his autobiography" (1981), published after his death thanks to his daughter Zofia Kuratowska, who prepared his notes for printing. Kazimierz Kuratowski represented Polish mathematics in the International Mathematics Union where he was vice president from 1963 to 1966. He also participated in numerous international congresses and lectured at dozens of universities around the world. He was an honorary causa doctor at the Universities in Glasgow, Prague, Wroclaw, and Paris. He received the highest national awards, as well as a gold medal of the Czechoslovak Academy of Sciences, and the Copernicus Polish Academy of Science award. Kuratowski died on June 18, 1980 in Warsaw.

A prize named after Kazimierz Kuratowski has been awarded annually to mathematicians up to 30 years of age since 1981.