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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] |
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] |
Michael Selection Theorem
In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: : Let ''X'' be a paracompact space and ''Y'' a Banach space. :Let F\colon X\to Y be a lower hemicontinuous set-valued function with nonempty convex closed values. :Then there exists a continuous selection f\colon X \to Y of ''F.'' : Conversely, if any lower semicontinuous multimap from topological space ''X'' to a Banach space, with nonempty convex closed values, admits a continuous selection, then ''X'' is paracompact. This provides another characterization for paracompactness. Examples A function that satisfies all requirements The function: F(x)= -x/2, ~1-x/4, shown by the grey area in the figure at the right, is a set-valued function from the real interval ,1to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: f(x)= 1-x/2 or f(x)= 1- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivalued Function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a codomain associates each in to one or more values in ; it is thus a serial binary relation. Some authors allow a multivalued function to have no value for some inputs (in this case a multivalued function is simply a binary relation). However, in some contexts such as in complex analysis (''X'' = ''Y'' = C), authors prefer to mimic function theory as they extend concepts of the ordinary (single-valued) functions. In this context, an ordinary function is often called a single-valued function to avoid confusion. The term ''multivalued function'' originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f(z) in some neighbourhood of a point z=a. This is the case fo ... [...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] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Choice Function
A choice function (selector, selection) is a mathematical function ''f'' that is defined on some collection ''X'' of nonempty sets and assigns some element of each set ''S'' in that collection to ''S'' by ''f''(''S''); ''f''(''S'') maps ''S'' to some element of ''S''. In other words, ''f'' is a choice function for ''X'' if and only if it belongs to the direct product of ''X''. An example Let ''X'' = . Then the function that assigns 7 to the set , 9 to , and 2 to is a choice function on ''X''. History and importance Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, some sets can still be shown to have a ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperspace (topology)
In the mathematical branch of topology, a hyperspace (or a space equipped with a hypertopology) is a topological space, which consists of the set ''CL(X)'' of all closed subsets of another topological space ''X'', equipped with a topology so that the canonical map i : x \mapsto \overline, is a homeomorphism onto its image. As a consequence, a copy of the original space ''X'' lives inside its hyperspace ''CL(X)''. Early examples of hypertopology include the Hausdorff metric and Vietoris topology. See also * Hausdorff distance * Kuratowski convergence * Wijsman convergence Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence. History The converge ... References External links Comparison of Hypertopologies [...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] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
