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Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
ian. He was one of the leading representatives of the
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 publish ...
. He worked as a professor at the
University of Warsaw The University of Warsaw (, ) is a public university, public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well ...
and at the Mathematical Institute of the
Polish Academy of Sciences The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars a ...
(IM PAN). Between 1946 and 1953, he served as President of the
Polish Mathematical Society The Polish Mathematical Society () is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). History The society was ...
. He is primarily known for his contributions to
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
,
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
,
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
. Some of the notable mathematical concepts bearing Kuratowski's name include
Kuratowski's theorem In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a Glossary of graph theory#Su ...
,
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a Set (mathematics), set. They are equivalent to the more commonly used open set definition. The ...
, Kuratowski-Zorn lemma and Kuratowski's intersection theorem.


Life and career


Early life

Kazimierz Kuratowski was born in
Warsaw Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at ...
, (then part of
Congress Poland Congress Poland or Congress Kingdom of Poland, formally known as the Kingdom of Poland, was a polity created in 1815 by the Congress of Vienna as a semi-autonomous Polish state, a successor to Napoleon's Duchy of Warsaw. It was established w ...
controlled by the
Russian Empire The Russian Empire was an empire that spanned most of northern Eurasia from its establishment in November 1721 until the proclamation of the Russian Republic in September 1917. At its height in the late 19th century, it covered about , roughl ...
), on 2 February 1896. He was a son of Marek Kuratow, a
barrister A barrister is a type of lawyer in common law jurisdiction (area), jurisdictions. Barristers mostly specialise in courtroom advocacy and litigation. Their tasks include arguing cases in courts and tribunals, drafting legal pleadings, jurisprud ...
, 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 The University of Glasgow (abbreviated as ''Glas.'' in Post-nominal letters, post-nominals; ) is a Public university, public research university in Glasgow, Scotland. Founded by papal bull in , it is the List of oldest universities in continuous ...
in
Scotland Scotland is a Countries of the United Kingdom, country that is part of the United Kingdom. It contains nearly one-third of the United Kingdom's land area, consisting of the northern part of the island of Great Britain and more than 790 adjac ...
, 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 or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
precluded any further enrolment. In 1915, Russian forces withdrew from Warsaw and
Warsaw University The University of Warsaw (, ) is a public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well as 100 specializat ...
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 the newly established
Second Polish Republic The Second Polish Republic, at the time officially known as the Republic of Poland, was a country in Central and Eastern Europe that existed between 7 October 1918 and 6 October 1939. The state was established in the final stage of World War I ...
.


Doctoral thesis

In autumn 1921 Kuratowski was awarded the Ph.D. degree for his groundbreaking work. His thesis statement consisted of two parts. One was devoted to an
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fi ...
construction of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
via the closure axioms. This first part (republished in a slightly modified form in 1922) has been cited in hundreds of scientific articles. The second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was
Stefan Mazurkiewicz Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). ...
. Kuratowski's thesis solved certain problems in
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, 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 Lviv ( or ; ; ; see #Names and symbols, below for other names) is the largest city in western Ukraine, as well as the List of cities in Ukraine, fifth-largest city in Ukraine, with a population of It serves as the administrative centre of ...
, 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 Society Warsaw Scientific Society (Polish: ''Towarzystwo Naukowe Warszawskie''; TNW) is a Polish scientific society based in Warsaw Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city o ...
While Kuratowski associated with many of the scholars of the
Lwów School of Mathematics The Lwów school of mathematics () was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, ...
, such as
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
and
Stanislaw Ulam Stanislav and variants may refer to: People *Stanislav (given name), a Slavic given name with many spelling variations (Stanislaus, Stanislas, Stanisław, etc.) Places * Stanislav, Kherson Oblast, a coastal village in Ukraine * Stanislaus County, ...
, 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, collaborate closely with Banach in solving important problems in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
. In 1934 he was appointed the professor at
Warsaw University The University of Warsaw (, ) is a public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well as 100 specializat ...
. A year later Kuratowski was nominated as the head of the 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 Warsaw, since higher education for Poles was forbidden under
German occupation German-occupied Europe, or Nazi-occupied Europe, refers to the sovereign countries of Europe which were wholly or partly militarily occupied and civil-occupied, including puppet states, by the (armed forces) and the government of Nazi Germany at ...
. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the
Polish Academy of Learning The Polish Academy of Arts and Sciences or Polish Academy of Learning (, PAU), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of sciences (the other being the Po ...
, 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 The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars a ...
, 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 Institute of Mathematics, 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 served as vice-president of the
International Mathematical Union The International Mathematical Union (IMU) is an international organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports the International ...
(1963–1966) as well as 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 (1 November 1913 – 22 August 1975) was a Polish mathematician. He worked primarily in logic and foundations of mathematics and is perhaps best remembered for the Mostowski collapse lemma. He was a member of the Polish Academy ...
.


Legacy

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 The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars a ...
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 Edinburgh is the capital city of Scotland and one of its 32 Council areas of Scotland, council areas. The city is located in southeast Scotland and is bounded to the north by the Firth of Forth and to the south by the Pentland Hills. Edinburgh ...
,
Austria Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Aust ...
,
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
,
Hungary Hungary is a landlocked country in Central Europe. Spanning much of the Pannonian Basin, Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia and ...
,
Italy Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
and the
Union of Soviet Socialist Republics The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a transcontinental country that spanned much of Eurasia from 1922 until it dissolved in 1991. During its existence, it was the largest country by are ...
(USSR).


Kazimierz Kuratowski Prize

In 1981, IMPAN, the
Polish Mathematical Society The Polish Mathematical Society () is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). History The society was ...
, and Kuratowski's daughter Zofia Kuratowska established a prize in his name, the Kuratowski Prize, for achievements in mathematics to people under the age of 30 years. The prize is considered the most prestigious of awards for young Polish mathematicians; past recipients have included Józef H. Przytycki, Mariusz Lemańczyk, Tomasz Łuczak, Mikołaj Bojańczyk, and Wojciech Samotij.


Research

Kuratowski's research mainly focused on abstract
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
and
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
structures. He implemented the closure axioms (known in mathematical circles as the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a Set (mathematics), set. They are equivalent to the more commonly used open set definition. The ...
). 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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s (theory), and also research in the field of cutting
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s. Together with Ulam, who was Kuratowski's most talented student during the
Lwów Lviv ( or ; ; ; see #Names and symbols, below for other names) is the largest city in western Ukraine, as well as the List of cities in Ukraine, fifth-largest city in Ukraine, with a population of It serves as the administrative centre of ...
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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, including research with
Banach Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase "Son of man (Judaism), son of man", combining the Hebrew language, Hebrew word ' ...
and Tarski, was continued by many students. Moreover, with
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
and
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...
he provided most of the theory concerning
Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
s (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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
) in 1922. This result has important connections to many basic theorems. Zorn gave its application in 1935. Kuratowski implemented many concepts in
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
and topology. In many cases, Kuratowski established new terminologies and symbolisms. His contributions to mathematics include: * a characterization of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s which are now called the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a Set (mathematics), set. They are equivalent to the more commonly used open set definition. The ...
; * proof of the Kuratowski–Zorn lemma; * in
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, the characterization of
planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
s now known as
Kuratowski's theorem In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a Glossary of graph theory#Su ...
; * identification of the
ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
(x,y) with the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
\; * the Kuratowski finite set definition, see Kuratowski-finite; * introduction of the
Tarski–Kuratowski algorithm In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm that produces an upper bound for the complexity of a given formula in the arithmetical hierarchy and analytical hierarchy. The algori ...
; *
Kuratowski's closure-complement problem In 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 an ...
; * Kuratowski's free set theorem; * Kuratowski's intersection theorem; * Knaster-Kuratowski fan; * Kuratowski-Ulam theorem; * Kuratowski convergence of subsets of metric spaces; * the
Kuratowski and Ryll-Nardzewski measurable selection theorem In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathema ...
; Kuratowski's post-war works were mainly focused on three strands: * The development of
homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
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 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). 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 Mathematical 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 The Czechoslovak Academy of Sciences (Czech: ''Československá akademie věd'', Slovak: ''Česko-slovenská akadémia vied'') was established in 1953 to be the scientific center for Czechoslovakia. It was succeeded by the Czech Academy of Science ...
, and the
Polish Academy of Sciences The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars a ...
. Kuratowski died on 18 June 1980 in Warsaw. *


See also

*
List of Polish mathematicians A list of notable Poland, Polish mathematicians: References

{{DEFAULTSORT:Polish mathematicians Polish mathematicians, Lists of Polish people by occupation, Mathematicians Lists of mathematicians by nationality, Polish ...
* Scottish Café * List of things named after Kazimierz Kuratowski *
Timeline of Polish science and technology Education has been of prime interest to Poland's rulers since the early 12th century. The catalog of the library of the Cathedral Chapter in Kraków dating from 1110 shows that Polish scholars already then had access to western European literature ...


Notes


References

* (in Polish) * * * * .


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. * * {{DEFAULTSORT:Kuratowski, Kazimierz 1896 births 1980 deaths 20th-century Polish philosophers Warsaw School of Mathematics People from Warsaw Governorate Fellows of the Royal Society of Edinburgh Foreign members of the USSR Academy of Sciences Members of the Polish Academy of Learning Members of the Polish Academy of Sciences Polish logicians Set theorists Topologists University of Warsaw alumni Academic staff of the University of Warsaw Institute for Advanced Study visiting scholars Members of the German Academy of Sciences at Berlin Recipients of the Medal of the 10th Anniversary of the People's Republic of Poland