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Rothberger Space
In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers \mathcal_1, \mathcal_2, \ldots of the space there are sets U_1 \in \mathcal_1, U_2 \in \mathcal_2, \ldots such that the family \ covers the space. History In 1938, Fritz Rothberger introduced his property known as C''. Characterizations Combinatorial characterization For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space \mathbb^\mathbb. A subset A of \mathbb^\mathbb is guessable if there is a function g\in A such that the sets \ are infinite for all functions f\in A. A subset of the real line is Rothberger iff every continuous image of that space into the Baire space is guessable. In particular, every subset of the real line of cardinality less than \mathrm(\mathcal) is Rothberger. Topological game characterization Le ... [...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] |
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, Cichoń'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] |
Taylor & Francis
Taylor & Francis Group is an international company originating in England that publishes books and academic journals. Its parts include Taylor & Francis, Routledge, F1000 Research or Dovepress. It is a division of Informa plc, a United Kingdom–based publisher and conference company. Overview The company was founded in 1852 when William Francis joined Richard Taylor in his publishing business. Taylor had founded his company in 1798. Their subjects covered agriculture, chemistry, education, engineering, geography, law, mathematics, medicine, and social sciences. Francis's son, Richard Taunton Francis (1883–1930), was sole partner in the firm from 1917 to 1930. In 1965, Taylor & Francis launched Wykeham Publications and began book publishing. T&F acquired Hemisphere Publishing in 1988, and the company was renamed Taylor & Francis Group to reflect the growing number of imprints. Taylor & Francis left the printing business in 1990, to concentrate on publishing. In 1998 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luzin Set
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrary number of isolated points. The existence of a Luzin space is independent of the axioms of ZFC. showed that the continuum hypothesis implies that a Luzin space exists. showed that assuming Martin's axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces. In real analysis In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as an uncountable subset of the reals such that every uncountable subset of is nonmeager; that is, of second Baire category. Equivalently, is an uncountable set of reals that meets every first category set in only countably many points. Luzin proved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Measure Zero Set
In mathematical analysis, a strong measure zero set is a subset ''A'' of the real line with the following property: :for every sequence (ε''n'') of positive reals there exists a sequence (''In'') of intervals such that , ''I''''n'', < ε''n'' for all ''n'' and ''A'' is contained in the union of the ''I''''n''. (Here , ''I''''n'', denotes the length of the interval ''I''''n''.) Every is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has 0. The is an exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laver Model
{{Disambiguation, geo ...
Laver may refer to: * Laver (surname), a list of people with the name * Laver (ghost town), Sweden * Green laver, a type of edible green seaweed used to make laverbread * River Laver, a river in North Yorkshire, England * Lavatorium, a washing facility in a monastery * A basin for ritual purification * Laver Bariu (1929–2014), Albanian folk clarinetist and singer See also * Leaver, a surname * Lever (other) A lever is a mechanical device to multiply force. Lever may also refer to: Entertainment and media * "Lever" (song), a 1998 single by The Mavis's * "The Lever" (song), a 2002 song by Silverchair *''The Lever'' (website), an American news outle ... [...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] |
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