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List Of Things Named After Kazimierz Kuratowski ...
This is a (partial) list of things named after Kazimierz Kuratowski, a 20th-century Polish mathematician and logician associated with the Warsaw School of Mathematics: Mathematics *Kuratowski's theorem *Kuratowski closure axioms *Kuratowski convergence * Kuratowski–Zorn lemma *Kuratowski's closure-complement problem * Kuratowski's intersection theorem *Kuratowski embedding * Kuratowski-Ulam theorem * Kuratowski-finite *Kuratowski and Ryll-Nardzewski measurable selection theorem *Knaster–Kuratowski–Mazurkiewicz lemma * Kuratowski's free set theorem *Knaster–Kuratowski fan Logic *Tarski–Kuratowski algorithm Other *Kuratowski Prize * 26205 Kuratowski, a minor planet See also *{{intitle, Kuratowski Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Mathematical Institute of the Polish Academy of Sciences (IM PAN). Between 1946 and 1953, he served as President of the Polish Mathematical Society. He is primarily known for his contributions to set theory, topology, measure theory and graph theory. Some of the notable mathematical concepts bearing Kuratowski's name include Kuratowski's theorem, Kuratowski closure axioms, Kuratowski-Zorn lemma and Kuratowski's intersection theorem. Life and career Early life Kazimierz Kuratowski was born in Warsaw, (then part of Congress Poland controlled by the Russian Empire), on 2 February 1896. 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. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski-finite
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an ''infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n is the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski Prize
The Kuratowski Prize (Polish: ''Nagroda im. Kazimierza Kuratowskiego'') is a Polish annual mathematics award conferred jointly by the Polish Academy of Sciences (PAN) and the Polish Mathematical Society (PTM) for contributions in the field of mathematics granted to individuals under the age of 30. It is named in honour of Polish mathematician and logician Kazimierz Kuratowski (1896–1980). Description and history The prize was established in 1981 on the initiative of physician and politician Zofia Kuratowska, who was personally the daughter of Kazimierz Kuratowski. It is presented annually by the Institute of Mathematics of the Polish Academy of Sciences and the Polish Mathematical Society (''Polskie Towarzystwo Matematyczne''). The Kuratowski Prize ceremony takes place during the scientific session of the Polish Mathematical Society and the laureate of the prize is invited to give a speech on a chosen subject. It is considered the most prestigious award for young mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 algorithm is named after Alfred Tarski and Kazimierz Kuratowski. Algorithm The Tarski–Kuratowski algorithm for the arithmetical hierarchy consists of the following steps: # Convert the formula to prenex normal form. (This is the non-deterministic part of the algorithm, as there may be more than one valid prenex normal form for the given formula.) # If the formula is quantifier-free, it is in \Sigma^0_0 and \Pi^0_0. # Otherwise, count the number of alternations of quantifiers; call this ''k''. # If the first quantifier is ∃, the formula is in \Sigma^0_. # If the first quantifier is ∀ A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knaster–Kuratowski Fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected. It is also known as Cantor's leaky tent or Cantor's teepee (after Georg Cantor), depending on the presence or absence of the apex. Let C be the Cantor set, let p be the point \left(\tfrac1,\tfrac1\right)\in\mathbb R^2, and let L(c), for c \in C, denote the line segment connecting (c,0) to p. If c \in C is an endpoint of an interval deleted in the Cantor set, let X_ = \; for all other points in C let X_ = \; the Knaster–Kuratowski fan is defined as \bigcup_ X_ equipped with the subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski's Free Set Theorem
Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It was largely forgotten for decades, but has been applied recently in solving several lattice theory problems, such as the congruence lattice problem. Denote by the set of all finite subsets of a set X. Likewise, for a positive integer n, denote by n the set of all n-elements subsets of X. For a mapping \Phi\colon n\to , we say that a subset U of X is ''free'' (with respect to \Phi), if for any n-element subset V of U and any u\in U\setminus V, u\notin\Phi(V). Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form \aleph_n. The theorem states the following. Let n be a positive integer and let X be a set. Then the cardinality of X is greater than or equal to \aleph_n if and only if for every mapping \Phi from n to , there exists an (n+1)-element free subset of X with respect to \Phi. For n=1, Kuratowski's free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knaster–Kuratowski–Mazurkiewicz Lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. Statement Let \Delta_ be an (n-1)-dimensional simplex with ''n'' vertices labeled as 1,\ldots,n. A KKM covering is defined as a set C_1,\ldots,C_n of closed sets such that for any I \subseteq \, the convex hull of the vertices corresponding to I is covered by \bigcup_C_i. The KKM lemma says that in every KKM covering, the common intersection of all ''n'' sets is nonempty, i.e.: :\bigcap_^n C_i \neq \emptyset. Example When n=3, the KKM lemma considers the simplex \Delta_2 which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets C_1,C_2,C_3 such that: * C_1 covers vertex 1, C_2 covers vertex 2, C_3 covers vertex 3. * The edge 12 (from vertex 1 to vertex 2) is covered by the set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski. Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control. Statement of the theorem Let X be a Polish space, \mathcal (X) the Borel σ-algebra of X , (\Omega, \mathcal) a measurable space and \psi a multifunction on \Omega taking values in the set of nonempty closed subsets of X . Suppose that \psi is \mathcal -weakly measurable, that is, for every open subset U of X , we have :\ \in \mathcal. Then \psi has a selection Selection may refer to: Science * Selection (biology), also called natural selection, selection in evolution ** Sex selection, in genetics ** Mate select ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in 1920—one of the world's first specialist pure-mathematics journals. It was in this journal, in 1933, that Alfred Tarski—whose illustrious career would a few years later take him to the University of California, Berkeley—published his celebrated theorem on the undefinability of the notion of truth. Notable members of the Warsaw School of Mathematics have included: * Wacław Sierpiński * Kazimierz Kuratowski * Edward Marczewski * Bronisław Knaster * Zygmunt Janiszewski * Stefan Mazurkiewicz * Stanisław Saks * Karol Borsuk * Roman Sikorski * Nachman Aronszajn * Samuel Eilenberg Additionally, notable logicians of the Lwów–Warsaw School of Logic, working at War ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski Embedding
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement obviously holds for the empty space. If (''X'',''d'') is a metric space, ''x''0 is a point in ''X'', and ''Cb''(''X'') denotes the Banach space of all bounded continuous real-valued functions on ''X'' with the supremum norm, then the map :\Phi : X \rarr C_b(X) defined by :\Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox\quad x,y\in X is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. The Kuratowski–Wojdysławski theorem states that every bounded metric space ''X'' is isometric to a closed subset of a convex subset of some Banach space. (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry :\Psi : X \rarr C_b(X) defined by :\Psi(x)(y) = d(x,y) \quad\mbox\quad x,y\in X The convex set menti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski's Intersection Theorem
In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be compact, Kuratowski's result allows them to be non-compact, but insists that their non-compactness "tends to zero" in an appropriate sense. The theorem is named for the Polish mathematician Kazimierz Kuratowski, who proved it in 1930. Statement of the theorem Let (''X'', ''d'') be a complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou .... Given a subset ''A'' ⊆ ''X'', its Kuratowski measure of non-compactness ''α''(''A'') ≥ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
