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Menger Space
In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open covers \mathcal_1, \mathcal_2, \ldots of the space there are finite sets \mathcal_1 \subset \mathcal_1, \mathcal_2 \subset \mathcal_2, \ldots such that the family \mathcal_1 \cup \mathcal_2 \cup \cdots covers the space. History In 1924, Karl Menger introduced the following basis property for metric spaces: Every basis of the topology contains a countable family of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz observed that Menger's basis property can be reformulated to the above form using sequences of open covers. Menger's conjecture Menger conjectured that in ZFC every Menger metric space is σ-compact. A. W. Miller and D. H. Fremlin proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some 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 mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selection Principle
In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. The main selection principles In 1924, Karl Menger introduced the following basis property for metric spaces: Every basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz observed that Menger's basis property is equivalent to the following selective property: for every sequence of op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-compact Space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and locally compact. Properties and examples * Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover). The reverse implications do not hold, for example, standard Euclidean space (R''n'') is σ-compact but not compact, and the lower limit topology on the real line is Lindelöf but not σ-compact. In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact. However, it is true that any locally compact Lindelöf space is σ-compact. *A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point. * If ''G'' is a topological group and ''G'' is locally compact at one point, then ''G'' is locally compact everywhere. Therefore, the previous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Menger
Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves and regions; in graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences. Biography Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and the Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology (IIT) in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boaz Tsaban
Boaz Tsaban (born February 1973) is an Israeli mathematician on the faculty of Bar-Ilan University. His research interests include selection principles within set theory and nonabelian cryptology, within mathematical cryptology. Biography Boaz Tsaban grew up in Or Yehuda, a city near Tel Aviv. At the age of 16 he was selected with other high school students to attend the first cycle of a special preparation program in mathematics, at Bar-Ilan University, being admitted to regular mathematics courses at the University a year later. He completed his B.Sc., M.Sc. and Ph.D. degrees with highest distinctions. Two years as a post-doctoral fellow at Hebrew University were followed by a three-year Koshland Fellowship at the Weizmann Institute of Science before he joined the Department of Mathematics, Bar-Ilan University in 2007. Academic career In the field of selection principles, Tsaban devised the method of omission of intervals for establishing covering properties of sets of real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, analysis, in particular functional analysis. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details. A topological space X is called a Baire space if it satisfies any of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cichoń's Diagram
In set theory, CichoÅ„'s diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to \aleph_1, the smallest uncountable cardinal, and they are bounded above by 2^, the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets (first category sets). Definitions Let ''I'' be an ideal of a fixed infinite set ''X'', containing all finite subsets of ''X''. We define the following " cardinal coefficients" of ''I'': *\operatorname(I)=\min\. ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least \aleph_0; if ''I'' is a σ-ideal, then add('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindelöf Space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' subcover. A hereditarily Lindelöf space is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term ''hereditarily Lindelöf'' is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. Properties of Lindelöf spaces * Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf. * A Lindelöf space is compact if and only if it is countably compact. * Every second-countable space is Lindelöf, but not conversely. For example, there are many compact sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathias Forcing
In mathematics, forcing is a method of constructing new models ''M'' 'G''of set theory by adding a generic subset ''G'' of a poset ''P'' to a model ''M''. The poset ''P'' used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable ''P''. This article lists some of the posets ''P'' that have been used in this construction. Notation *''P'' is a poset with order < *''V'' is the universe of all sets *''M'' is a countable transitive model of set theory *''G'' is a generic subset of ''P'' over ''M''. Definitions *''P'' satisfies the if every antichain in ''P'' is at most countable. This implies that ''V'' and ''V'' 'G''have the sam ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Topological Spaces
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object * Material properties, properties by which the benefits of one material versus another can be assessed * Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system * Semantic property * Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances * Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as nam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
