Kazimierz Kuratowski

Kazimierz Kuratowski was born in Warsaw, Vistula Land (the part of the former Kingdom of Poland controlled by the Russian Empire), on 2 February 1896, into an assimilated Jewish family. He was a son of Marek Kuratow, a barrister, and Róża Karzewska. 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; instruction in Polish was prohibited. He completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics. He obtained his Ph.D. in 1921, in newly independent Poland.

Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów'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 Lwów’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, including 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 mathematical circles as the Kuratowski closure axioms). This was fundamental for the development of topological space theory and irreducible continuum theory between two points. The most valuable results, which were obtained by 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 Lwów 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 and 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 and Kuratowski brought a comprehensive and precise study to connected components theory. It was applied to issues such as cutting-plane, with the paradoxical examples of connected components.

Kuratowski proved the Kuratowski-Zorn lemma (often called just Zorn's lemma) in 1922.[5] This result has important connections to many basic theorems. Zorn gave its application in 1935.[6] Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminologies and symbolisms. His contributions to mathematics include:

• a characterization of topological spaces which are now called the 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 ${\displaystyle (x,y)}$ with the set ${\displaystyle \{\{x\},\{x,y\}\};}$[7]
• the Kuratowski finite set definition, see Kuratowski-finite;[8]
• introduction of the Tarski–Kuratowski algorithm;
• Kuratowski's closure-complement problem;
• Kuratowski's free set theorem;
• Knaster-Kuratowski fan;
• Kuratowski-Ulam theorem;
• Kuratowski convergence of subsets of metric spaces;
• the Kuratowski and 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 Topologie (Vol. I, 1933, translated into English and Russian, and Vol. II, 1950) and Introduction to Set Theory and Topology (Vol. I, 1952, translated into English, French, Spanish, and Bulgarian). He authored "A Half Century of Polish Mathematics 1920-1970: Remembrances and Reflections" (1973)[9] and "Notes to his autobiography" (1981). The latter was published posthumously thanks to Kuratowski's 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. What is more, he 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 Polish Academy of Science. Kuratowski died on 18 June 1980 in Warsaw.

• Kuratowski, Kazimierz; Mostowski, Andrzej (1976) [1968], Set theory. With an introduction to descriptive set theory, Studies in Logic and the Foundations of Mathematics, 86 (Second ed.), Amsterdam-New York-Oxford: North-Holland Publishing Co., MR 0485384

• List of Polish mathematicians
• Scottish Café

• TOPOLOGIE I, Espaces Métrisables, Espaces Complets Monografie Matematyczne series, vol. 20, Polish Mathematical Society, Warszawa-Lwów, 1948.
• TOPOLOGIE II, Espaces Compacts, Espaces Connexes, Plan Euclidien Monografie Matematyczne series, vol. 21, Polish Mathematical Society, Warszawa-Lwów, 1950.
• O'Connor, John J.; Robertson, Edmund F., "Kazimierz Kuratowski", MacTutor History of Mathematics archive, University of St Andrews.
• Kazimierz Kuratowski at the Mathematics Genealogy Project