Kazimierz Kuratowski
Encyclopedia
Kazimierz Kuratowski was a Polish
mathematician
and logician. He was one of the leading representatives of the Warsaw School of Mathematics
.
, at the time part of Poland controlled by Tsarist Russia, on February 2, 1896. He was a son of Marek Kuratow, a barrister
, and Róża Karzewski. 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
.
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.
While Kuratowski associated with many of the scholars of the Lwów School of Mathematics
, such as Stefan Banach
and Stanislaw Ulam, and the circle of mathematicians based around the Scottish Café
he kept close connections with Warsaw. Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book
was begun (in 1935), hence did not contribute any problems to it. He did however, write collaborate closely with Banach in solving important problems in measure theory.
In 1934 he was appointed the professor at Warsaw University. A year later Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences.
, since higher education for Poles was forbidden under German occupation.
In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning
, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, and from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences
, of which 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. He helped to establish the State Mathematical Institute, which was incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 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 (1968-1980). From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski
.
. 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).
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 (often called just a Zorn's 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:
Kuratowski’s post-war works were mainly focused on three strands:
, Edition I, 1952, translation into English
and Russian
), and is an introduction to set theory
and topology
(edition I, 1952, translated into English, French, Spanish, and Bulgarian). He was a founder of studying 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). 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 Copernicus Polish Academy of Science awarded. Kuratowski died on June 18, 1980 in Warsaw.
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and logician. He was one of the leading representatives of the Warsaw School of Mathematics
Warsaw School of Mathematics
"Warsaw School of Mathematics" is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They published in the journal Fundamenta Mathematicae, founded...
.
Biography and studies
Kazimierz Kuratowski was born in WarsawWarsaw
Warsaw is the capital and largest city of Poland. It is located on the Vistula River, roughly from the Baltic Sea and from the Carpathian Mountains. Its population in 2010 was estimated at 1,716,855 residents with a greater metropolitan area of 2,631,902 residents, making Warsaw the 10th most...
, at the time part of Poland controlled by Tsarist Russia, on February 2, 1896. He was a son of Marek Kuratow, a barrister
Barrister
A barrister is a member of one of the two classes of lawyer found in many common law jurisdictions with split legal professions. Barristers specialise in courtroom advocacy, drafting legal pleadings and giving expert legal opinions...
, and Róża Karzewski. 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
University of Glasgow
The University of Glasgow is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities. Located in Glasgow, the university was founded in 1451 and is presently one of seventeen British higher education institutions ranked amongst the top 100 of the...
in Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
, 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
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
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
Second Polish Republic
The Second Polish Republic, Second Commonwealth of Poland or interwar Poland refers to Poland between the two world wars; a period in Polish history in which Poland was restored as an independent state. Officially known as the Republic of Poland or the Commonwealth of Poland , the Polish state was...
.
Doctoral thesis
His thesis statement consisted of two parts. One was devoted to an axiomaticAxiomatic
* In mathematics, an "axiomatic" theory is one based on axioms* Axiomatic , a collection of short stories by Greg Egan* Axiomatic , a 2005 album by Australian band Taxiride...
depiction of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
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
Zygmunt Janiszewski
Zygmunt Janiszewski was a Polish mathematician.-Life:His mother was Julia Szulc-Chojnicka. His father, Czeslaw Janiszewski, was a graduate of the University of Warsaw and was an important person in finance, being the director of the Société du Crédit Municipal in Warsaw.Janiszewski taught at the...
. 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
Charles Jean de la Vallée-Poussin
Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...
, Baron de la Vallée Poussin.
Academic career until World War II
Two years later, in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University. He was then appointed a full professor of mathematics at Lwów Polytechnic in Lwów (now Lviv, Ukraine), in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was also dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific SocietyWarsaw Scientific Society
Warsaw Scientific Society is a Polish scientific society based in Warsaw. It was established in 1907 as a continuation of the Society of Friends of Science to advance the sciences and arts and to publish scientific papers....
While Kuratowski associated with many of the scholars of the Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...
, such as Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
and Stanislaw Ulam, and the circle of mathematicians based around the Scottish Café
Scottish Café
The Scottish Café was the café in Lwów where, in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in functional analysis and topology....
he kept close connections with Warsaw. Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book
Scottish Book
The Scottish Book was a thick notebook used by mathematicians of the Lwow School of Mathematics for jotting down problems meant to be solved. The notebook was named after the "Scottish Café" where it was kept....
was begun (in 1935), hence did not contribute any problems to it. He did however, write collaborate closely with Banach in solving important problems in measure theory.
In 1934 he was appointed the professor at Warsaw University. A year later Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences.
During and after the war
During World War II, he gave lectures at the underground university in WarsawEducation in Poland during World War II
This article covers the topic of underground education in Poland during World War II. Secret learning prepared new cadres for the post-war reconstruction of Poland and countered the German and Soviet threat to exterminate the Polish culture....
, since higher education for Poles was forbidden under German occupation.
In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning
Polish Academy of Learning
The Polish Academy of Arts and Sciences or Polish Academy of Learning , headquartered in Kraków, is one of two institutions in contemporary Poland having the nature of an academy of sciences....
, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, and from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences
Polish Academy of Sciences
The Polish Academy of Sciences, headquartered in Warsaw, is one of two Polish institutions having the nature of an academy of sciences.-History:...
, of which 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. He helped to establish the State Mathematical Institute, which was incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 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 (1968-1980). From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski
Andrzej Mostowski
Andrzej Mostowski was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma....
.
Legacy
Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lviv's Scottish CaféScottish Café
The Scottish Café was the café in Lwów where, in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in functional analysis and topology....
. 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
Polish Academy of Sciences
The Polish Academy of Sciences, headquartered in Warsaw, is one of two Polish institutions having the nature of an academy of sciences.-History:...
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
Edinburgh
Edinburgh is the capital city of Scotland, the second largest city in Scotland, and the eighth most populous in the United Kingdom. The City of Edinburgh Council governs one of Scotland's 32 local government council areas. The council area includes urban Edinburgh and a rural area...
, Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
, Italy
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
and the Union of Soviet Socialist Republics (USSR).
Research
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 topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
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
Lviv
Lviv is a city in western Ukraine. The city is regarded as one of the main cultural centres of today's Ukraine and historically has also been a major Polish and Jewish cultural center, as Poles and Jews were the two main ethnicities of the city until the outbreak of World War II and the following...
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
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
and Wacław Sierpiński he provided most of the theory concerning Polish spaces
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
(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 (often called just a Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
), which was proven for the first time in 1922 by Kuratowski (Fundamenta Mathematicae
Fundamenta Mathematicae
Fundamenta Mathematicae is a scientific journal of mathematics with a special focus on the foundations of mathematics. At present, it concentrates on papers devoted to set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems...
, 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 spaceHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s which are now called Kuratowski closure axioms; - proof of the Kuratowski-Zorn lemma;
- in Graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, the characterization of Planar graphs now known as Kuratowski's theorem; - identification of the ordered pair (x,y) with the set ;
- introduction of the Tarski-Kuratowski algorithm;
- Kuratowski's closure-complement problemKuratowski's closure-complement problemIn point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by...
; - 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.
Publications
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 contens set theory (together with Andrzej MostowskiAndrzej Mostowski
Andrzej Mostowski was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma....
, Edition I, 1952, translation into English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
and Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
), and is an introduction to set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
(edition I, 1952, translated into English, French, Spanish, and Bulgarian). He was a founder of studying 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). 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
Czechoslovak Academy of Sciences
The Czechoslovak Academy of Sciences was established in 1953 to be the scientific center for Czechoslovakia. It was succeeded by the Academy of Sciences of the Czech Republic in 1992.-History:...
, and the Copernicus Polish Academy of Science awarded. Kuratowski died on June 18, 1980 in Warsaw.
External links
- 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.