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P-space
In the mathematical field of topology, there are various notions of a ''P''-space and of a ''p''-space. Generic use The expression ''P-space'' might be used generically to denote a topological space satisfying some given and previously introduced topological invariant ''P''. This might apply also to spaces of a different kind, i.e. non-topological spaces with additional structure. ''P-spaces'' in the sense of Gillman–Henriksen A ''P-space'' in the sense of Gillman– Henriksen is a topological space in which every countable intersection of open sets is open. An equivalent condition is that countable unions of closed sets are closed. In other words, Gδ sets are open and Fσ sets are closed. The letter ''P'' stands for both ''pseudo-discrete'' and ''prime''. Gillman and Henriksen also define a ''P-point'' as a point at which any prime ideal of the ring of real-valued continuous functions is maximal, and a P-space is a space in which every point is a P-point. Different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morita Conjectures
The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjectures, formulated by Kiiti Morita in 1976, asked # If X \times Y is normal for every normal space ''Y'', is ''X'' a discrete space? # If X \times Y is normal for every normal P-space#P-spaces in the sense of Morita, P-space ''Y'', is ''X'' metrizable? # If X \times Y is normal for every normal countably paracompact space ''Y'', is ''X'' metrizable and sigma-locally compact? The answers were believed to be affirmative. Here a normal P-space ''Y'' is characterised by the property that the product with every metrizable ''X'' is normal; thus the conjecture was that the converse holds. Keiko Chiba, Teodor C. Przymusiński, and Mary Ellen Rudin proved conjecture (1) and showed that conjectures (2) and (3) cannot be proven false under the standard Zermelo–Fraenkel set theory, ZFC axioms for mathematics (specifically, that the conjectures hold under the axiom o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Gillman
Leonard E. Gillman (January 8, 1917 – April 7, 2009) was an American mathematician, emeritus professor at the University of Texas at Austin. He was also an accomplished classical pianist. Biography Early life and education Gillman was born in Cleveland, Ohio in 1917. His family moved to Pittsburgh, Pennsylvania in 1922. It was there that he started taking piano lessons at age six. They moved to New York City in 1926, and he began intensive training as a pianist. Upon graduation from high school in 1933, Gillman won a fellowship to the Juilliard Graduate School of Music. Career After one semester at Juilliard, he enrolled in evening classes in French and mathematics at Columbia University. He received a diploma in piano from Juilliard in 1938, then continued his studies at Columbia, graduating with a B.S. in mathematics in 1941. He stayed on as a graduate student, and completed the coursework for a mathematics Ph.D. by 1943. In 1943, Gillman accepted a position at Tufts C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov-discrete Space
In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder ≤ on a set ''X'', there is a unique Alexandrov topology on ''X'' for which the specialization preorder is ≤. The open sets are just the upper sets with respect to ≤. Thus, Alexandrov topologies on ''X'' are in one-to-one correspondence with preorders on ''X''. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gδ Set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with ''G'' for '' Gebiet'' (''German'': area, or neighbourhood) meaning open set in this case and for '' Durchschnitt'' (''German'': intersection).. Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore. Gδ sets, and their dual, F sets, are the second level of the Borel hierarchy. Definition In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level Π sets of the Borel hierarchy. Examples * Any open set is trivially a Gδ set. * The irrational numbers are a Gδ set in the real numbers \R. They can be written as the countable intersection of the open sets \^ (the superscript denoting the complement) where q is rational. * The set of rational numbers \Q is a Gδ set in \R. If \Q were the intersection of open set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fσ Set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for (''French'': closed) and σ for (''French'': sum, union).. The complement of an Fσ set is a Gδ set. Fσ is the same as \mathbf^0_2 in the Borel hierarchy. Examples Each closed set is an Fσ set. The set \mathbb of rationals is an Fσ set in \mathbb. More generally, any countable set in a T1 space is an Fσ set, because every singleton \ is closed. The set \mathbb\setminus\mathbb of irrationals is not a Fσ set. In metrizable spaces, every open set is an Fσ set.. The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set. The set A of all points (x,y) in the Cartesian plane such that x/y is rational is an Fσ set because it can be expressed as the union of all the lines passing through the origin with rational slope: : A = \bigcup_ \, where \mathbb, is the set of rational number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Arhangelskii
Alexander Vladimirovich Arhangelskii (russian: Александр Владимирович Архангельский, ''Aleksandr Vladimirovich Arkhangelsky'', born 13 March 1938 in Moscow) is a Russian mathematician. His research, comprising over 200 published papers, covers various subfields of general topology. He has done particularly important work in metrizability theory and generalized metric spaces, cardinal functions, topological function spaces and other topological groups, and special classes of topological maps. After a long and distinguished career at Moscow State University, he moved to the United States in the 1990s. In 1993 he joined the faculty of Ohio University, from which he retired in 2011. Biography Arhangelskii was the son of Vladimir Alexandrovich Arhangelskii and Maria Pavlova Radimova, who divorced by the time he was four years old. He was raised in Moscow by his father. He was also close to his uncle, childless aircraft designer Alexander Arkhangelsky. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiiti Morita
was a Japanese mathematician working in algebra and topology. Morita was born in 1915 in Hamamatsu, Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo University of Science. He received his Ph.D. from Osaka University in 1950, with a thesis in topology. After teaching at the Tokyo Higher Normal School, he became professor at the University of Tsukuba in 1951. He held this position until 1978, after which he taught at Sophia University. Morita died of heart failure in 1995 at the Sakakibara Heart Institute in Tokyo; he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson. He introduced the concepts now known as Morita equivalence and Morita duality which were given wide circulation in the 1960s by Hyman Bass in a series of lectures. The Morita conjectures The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Topological Space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Topologies on a finite set Let X be a finite set. A topology on X is a subset \tau of P(X) (the power set of X ) such that # \varnothing \in \tau and X\in \tau . # if U, V \in \tau then U \cup V \in \tau . # if U, V \in \tau then U \cap V \in \tau . In other words, a subset \tau of P(X) is a topology if \tau contains both \varnothing and X and is closed under finite intersections and arbitrary unions. Elements of \tau are called open sets. Since the power set of a finite s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 space is locally finite if and only if it is discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a .... References * General topology Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
