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Kuratowski Convergence
In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902,This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets' text. the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets " accumulate". Definitions For a given sequence \_^ of points in a space X, a limit point of the sequence can be understood as any point x \in X where the sequence ''eventually'' becomes arbitrarily close to x. On the other hand, a cluster point of the sequence can be thought of as a point x \in X where the sequence ''frequently'' becomes arbitrarily close to x. The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space X. Metric Spaces Let (X,d) be a metric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Hemicontinuous
In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets ''A'' and ''B''. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions. Roughly speaking, a function is upper hemicontinuous if when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point. Upper hemicontinuity A correspondence \Gamma : A \to B is said t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Hemicontinuous
In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets ''A'' and ''B''. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions. Roughly speaking, a function is upper hemicontinuous if when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point. Upper hemicontinuity A correspondence \Gamma : A \to B is said ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Berge
Claude Jacques Berge (5 June 1926 – 30 June 2002) was a French mathematician, recognized as one of the modern founders of combinatorics and graph theory. Biography and professional history Claude Berge's parents were André Berge and Geneviève Fourcade. André Berge (1902–1995) was a physician and psychoanalyst who, in addition to his professional work, had published several novels. He was the son of the René Berge, a mining engineer, and Antoinette Faure. Félix François Faure (1841–1899) was Antoinette Faure's father; he was President of France from 1895 to 1899. André Berge married Geneviève in 1924, and Claude was the second of their six children. His five siblings were Nicole (the eldest), Antoine, Philippe, Edith, and Patrick. Claude attended the near Verneuil-sur-Avre, about west of Paris. This famous private school, founded by the sociologist Edmond Demolins in 1899, attracted students from all over France to its innovative educational program. At this stage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets ''A'' and ''B''. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions. Roughly speaking, a function is upper hemicontinuous if when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point. Upper hemicontinuity A correspondence \Gamma : A \to B is said ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vietoris Topology
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 topological sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Metric
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book ''Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice Fréchet in 1906, in his study of the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest subset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a ''positive scalar''. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. In this article, only the case of scalars in an ordered field is considered. Definition A subset ''C'' of a vector space ''V'' over an ordered field ''F'' is a cone (or sometimes called a linear cone) if for each ''x'' in ''C'' and positive scalar ''α'' in ''F'', the product ''αx'' is in ''C''. Note that some authors define co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also Continuous function, continuous functions. Such a topology is called a and every topological vector space has a Uniform space, uniform topological structure, allowing a notion of uniform convergence and Complete topological vector space, completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
