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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. Cited in ... [...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 ''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 ZFC axioms for mathematics (specifically, that the conjectures hold under the axiom of constructibility ''V=L''). Fifteen years later, Zoltán Tibor B ... [...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 Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ... at Columbia University. He received a diploma in piano from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov-discrete Space
In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov topology is one whose open sets are the upper sets for some preorder on the space. Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name stems from the fact that their topology is uniquely determined by the family of all finite subspaces. This makes them a generalization of finite topological spaces. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with Alexandrov spaces from Riemannian geometry introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. Characterizations of Alexandrov topologies Alexandrov topologies have numerous characterizations. In a topological space X, the following co ... [...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 (set theory), intersection of open sets. The notation originated from the German language, German nouns and . Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore. Gδ sets, and their dual, Fσ set, F sets, are the second level of the Borel hierarchy. Definition In a topological space a Gδ set is a countable intersection (set theory), 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 (set theory), complement) where q is Rational number, rational. * The set of rational numbers \Q is a Gδ set in \R. If \Q were the intersection of open sets A_n each ... [...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 an 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, deformations, such as Stretch factor, stretching, Torsion (mechanics), 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 List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the Zariski topology, is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal ''A'' is not necessarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Arhangelskii
Alexander Vladimirovich Arhangelskii (, ''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. In 1954, Arhangelskii entered Moscow State University, where he became a stud ... [...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 Tokyo, officially the Tokyo Metropolis, is the capital of Japan, capital and List of cities in Japan, most populous city in Japan. With a population of over 14 million in the city proper in 2023, it is List of largest cities, one of the most ...; he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson. He introduced the concepts now known as Morita equivalenc ... [...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 on ... [...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 set (mathematics), point set is finite set, 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 (structure), 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 arbitrary union (set theory), unions and intersection (set theory), ... [...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, a neighborhood consisting of finitely many elements. Background The conditions for local finiteness were created by Jun-iti Nagata and Yury Smirnov while searching for a stronger version of the Urysohn metrization theorem. The motivation behind local finiteness was to formulate a new way to determine if a topological space X is metrizable without the countable basis requirement from Urysohn's theorem. Definitions Let T = ( S, \tau ) be a topological space and let \mathcal be a set of subsets of S Then \mathcal is locally finite if and only if each element of S has a neighborhood which intersects a finite number of sets in \mathcal . A locally finite space is an Alexandrov space. 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 quantu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |