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Dowker Space
In the mathematics, mathematical field of general topology, a Dowker space is a topological space that is normal space, T4 but not paracompact space, countably paracompact. They are named after Clifford Hugh Dowker. The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as mathematical objects) helped mathematicians better understand the nature and variety of topological spaces. Equivalences Dowker showed, in 1951, the following: If ''X'' is a normal T1 space, T1 space (that is, a T4 space, T4 space), then the following are equivalent: * ''X'' is a Dowker space * The product of ''X'' with the unit interval is not normal. * ''X'' is not Metacompact space, countably metacompact. Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until Mary Ellen Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality \aleph_\omega^). Zoltán Tibor Balogh, Zoltán Balogh gave ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. The first specialized journal in the field of mathematics, originally it covered only topology, set theory, and foundations of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences. History The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland.According to and to the introduction to the 100th volume of the journal (1978, pp=1–2). These two sources cite an article written by Janiszewski himself in 1918 and titled "''On the needs of Mathematics in Poland''". Janiszewski posited that, to achieve its goal, the journal should not compel Polish mathematicians to submit articles written exclusively in Polish, and should be devoted only to a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saharon Shelah
Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Hebrew poet and Canaanist political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PCF Theory
PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities". Main definitions If ''A'' is an infinite set of regular cardinals, ''D'' is an ultrafilter on ''A'', then we let \operatorname\left(\prod A/D\right) denote the cofinality of the ordered set of functions \prod A where the ordering is defined as follows: f Main results Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. In analysis A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. Such examples were deemed pathological when they were first discovered. To quote Henri Poincaré: Since Poincaré, nowhere diffe ... [...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] |
Cardinality Of The Continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \bold, \bold\mathbb R\bold, . The real numbers \mathbb R are more numerous than the natural numbers \mathbb N. Moreover, \mathbb R has the same number of elements as the power set of \mathbb N. Symbolically, if the cardinality of \mathbb N is denoted as \aleph_0, the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers ''a'' < ''b'', no matter how close they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoltán Tibor Balogh
Zoltán "Zoli" Tibor Balogh (December 7, 1953 – June 19, 2002) was a Hungarian-born mathematician, specializing in set-theoretic topology. His father, Tibor Balogh, was also a mathematician. His best-known work concerned solutions to problems involving normality of products, most notably the first ZFC construction of a small ( cardinality continuum) Dowker space. He also solved Nagami's problem ( normal + screenable does not imply paracompact), and the second and third Morita conjectures about normality in products.Z. Balogh, Non-shrinking open covers and K. Morita's duality conjectures, '' Topology Appl.'', 115 (2001) 333-341 References External linksMemorial with photograph [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mary Ellen Rudin
Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young researcher, mainly in fields adjacent to general topology. Early life and education Mary Ellen (Estill) Rudin was born in Hillsboro, Texas to Joe Jefferson Estill and Irene (Shook) Estill. Her mother Irene was an English teacher before marriage, and her father Joe was a civil engineer. The family moved with her father's work, but spent a great deal of Mary Ellen's childhood around Leakey, Texas.Albers, D.J. and Reid, C. (1988) "An Interview with Mary Ellen Rudin". ''The College of Mathematics Journal'' 19(2) pp.114-137 She had one sibling, a younger brother. Both of Rudin's maternal grandmothers had attended Mary Sharp College near their hometown of Winchester, Tennessee. Rudin remarks on this legacy and how much her family valued educ ... [...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] |
Metacompact Space
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement. Properties The following can be said about metacompactness in relation to other properties of topological spaces: * Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank. * Every metacompact space is orthocompact. * Every metacompact normal space is a shrinking space * The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma. * An e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
