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Katětov–Tong Insertion Theorem
The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following: Let X be a normal topological space and let g, h\colon X \to \mathbb be functions with g upper semicontinuous, h lower semicontinuous, and g \leq h. Then there exists a continuous function f\colon X \to \mathbb with g \leq f \leq h. This theorem has a number of applications and is the first of many classical insertion theorems. In particular it implies the Tietze extension theorem and consequently Urysohn's lemma In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct contin ..., and so the conclusion of the theorem is equivalent to normality. References * {{DEFAULTSORT:Katetov-Tong insertion theorem General topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point-set Topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miroslav Katětov
Miroslav Katětov (; March 17, 1918, Chembar, Russia – December 15, 1995) was a Czech mathematician, chess master, and psychologist. His research interests in mathematics included topology and functional analysis. He was an author of the Katětov–Tong insertion theorem. From 1953 to 1957 he was rector of Charles University in Prague Charles University (CUNI; , UK; ; ), or historically as the University of Prague (), is the largest university in the Czech Republic. It is one of the oldest universities in the world in continuous operation, the oldest university north of the .... External links Biography* 1918 births 1995 deaths People from Penza Oblast Czechoslovak mathematicians Topologists Czech chess players Czech psychologists Charles University alumni Rectors of Charles University Czech expatriates in Russia 20th-century chess players 20th-century psychologists Soviet emigrants to Czechoslovakia {{europe-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hing Tong
Hing Tong (16 February 1922 – 4 March 2007) was an American mathematician. He is well known for providing the original proof of the Katetov–Tong insertion theorem. Life Hing Tong was born in Guangzhou, Canton, China. He received his bachelor's degree from the University of Pennsylvania. In 1947, he received his doctorate in mathematics from Columbia University, where his thesis advisor was Edgar Lorch. In 1956, he married fellow mathematician, Mary Powderly. He was the father of five children. Work Hing Tong made many significant contributions to the area of algebraic topology, and served in a number of academic capacities. In 1947, after receiving a National Research Council fellowship, he became an assistant professor at Barnard College (Columbia University). In 1955, he was a visiting scholar at the Institute for Advanced Study in Princeton. Also in 1955, he was appointed professor of mathematics (and eventually chairman of the mathematics department) at Wesleyan University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Space
Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * Normal (2009 film), ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * ''Normal!'', a 2011 Algerian film * The Normals (film), ''The Normals'' (film), a 2012 American comedy film * Normal (New Girl), "Normal" (''New Girl''), an episode of the TV series Mathematics * Normal (geometry), an object such as a line or vector that is perpendicular to a given object * Normal basis (of a Galois extension), used heavily in cryptography * Normal bundle * Normal cone, of a subscheme in algebraic geometry * Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) * Normal distribution, the Gaussian continuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicontinuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right). Briefly, a function on a domain X is lower semi-continuous if its epigraph \ is closed in X\times\R, and upper semi-continuous if -f is lower semi-continuous. A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x_0 to f\left(x_0\right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to f\left(x_0\right) - c then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tietze Extension Theorem
In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal 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 ... can be extended to the entire space, preserving boundedness if necessary. Formal statement If X is a normal space and f : A \to \R is a continuous map from a closed subset A of X into the real numbers \R carrying the standard topology, then there exists a of f to X; that is, there exists a map F : X \to \R continuous on all of X with F(a) = f(a) for all a \in A. Moreover, F may be chosen such that \sup \ ~=~ \sup \, that is, if f is bounded then F may be chosen to be bounded (with the same bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urysohn's Lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalised by (and usually used in the proof of) the Tietze extension theorem. The lemma is named after the mathematician Pavel Samuilovich Urysohn. Discussion Two subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are disjoint. In particular A and B are necessarily disjoint. Two plain subsets A and B are said to be separated by a continuous function if there exists a continuous function f : X \to , 1/math> from X into the unit interval , 1/math> such that f(a) = 0 for all a \in A and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
