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Phragmen–Brouwer Theorem
In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if ''X'' is a normal connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ... locally connected topological space, then the following two properties are equivalent: *If ''A'' and ''B'' are disjoint closed subsets whose union separates ''X'', then either ''A'' or ''B'' separates ''X''. *''X'' is unicoherent, meaning that if ''X'' is the union of two closed connected subsets, then their intersection is connected or empty. The theorem remains true with the weaker condition that ''A'' and ''B'' be separated. References * * * * García-Maynez, A. and Illanes, A. ‘A survey of multicoherence’, An. Inst. Autonoma Mexico 29 (1989) 17–67. * * Wilder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lars Edvard Phragmén
Lars Edvard Phragmén (2 September 1863, Örebro – 14 March 1937) was a Swedish mathematician who made contributions to complex analysis, voting theory, and actuarial science. He succeeded Sofia Kovalevskaia as professor of mathematical analysis at Stockholm University in 1892, where his research culminated in the development of the Phragmén–Lindelöf principle, and later served as president of the board of the Mittag-Leffler Institute. His pioneering "load-balancing" voting methods for proportional representation have experienced renewed interest in modern social choice theory and found practical application in Swedish parliamentary elections. Early life and career He was the son of a college professor and studied at Uppsala University and Stockholm University, graduating from Uppsala in 1889. He succeeded Sofia Kovalevskaia as professor of mathematical analysis at Stockholm University in 1892, and in 1884 provided a new proof of the Cantor–Bendixson theorem. His resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension. Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. In addition to his mathematical work, Brouwer also published the short philosophical tract ''Life, Art, and Mysticism'' (1905). Biography B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Topological Space
Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * '' Normal!'', a 2011 Algerian film * ''The Normals'' (film), a 2012 American comedy film * "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 continuous probability distribution * Normal equations, describing the solution of the linear least s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Topological Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topological space X the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected Topological Space
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicoherent
In mathematics, a unicoherent space is a topological space X that is connected and in which the following property holds: For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected. For example, any closed interval on the real line is unicoherent, but a circle is not. If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if ''X'' is a normal connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kon ... states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space. References * External links * Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Sets
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept. Definitions There are various ways in which two subsets A and B of a topological space X can be considered to be separated. A most basic way in which two sets can be separated is if they are disjoint, that is, if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. Each of the following properties is stricter than disjoint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Topology
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasonin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
