Uniform Space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closen ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] picture info Mathematical 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]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Functional Analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Metrizable Space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] R0 Space In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Symmetric Topology In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] picture info Tychonoff Space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps. Definitions A topological space X is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set A \subseteq X and any point x \in X \setminus A, there exists a real-valued continuous function f : X \to \R suc ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] picture info Hausdorff Space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separat ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Kolmogorov Space In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable. This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T1) is T0; this includes the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguisha ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] picture info Completely Regular Space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps. Definitions A topological space X is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set A \subseteq X and any point x \in X \setminus A, there exists a real-valued continuous function f : X \to \R suc ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Star Refinement In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''. The general definition makes sense for arbitrary coverings and does not require a topology. Let X be a set and let \mathcal U be a covering of X, i.e., X = \bigcup \mathcal U. Given a subset S of X then the ''star'' of S with respect to \mathcal U is the union of all the sets U\in \mathcal U that intersect S, i.e.: : \operatorname(S, \mathcal U) = \bigcup\big\. Given a point x\in X, we write \operatorname(x,\mathcal U) instead of \operatorname(\, \mathcal U). Note that \operatorname(S, \mathcal U) \ne \bigcup_ (O \cap S). The covering \mathcal U of X is said to be a ''refinement'' of a covering \mathcal V of X if every U\in \mathcal U is contained in some V\in \mathcal V. The covering \mathcal U is said to be a ''barycentric refinement'' of \mathcal V if for every x\in X the star \operatorname(x,\math ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Filter On A Partially Ordered Set Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that performs some operation on a multimedia stream * Email filtering, the processing of email to organize it according to specified criteria * Content-control software also known as an Internet filter * Wordfilter, a script typically used on Internet forums or chat rooms * Berkeley Packet Filter, filter expression used in the qualification of network data * DSL filter, a low-pass filter installed between analog devices and a telephone line * Helicon Filter, a raster graphics editor * Filter (large eddy simulation), a mathematical operation intended to remove a range of small scales from the solution to the Navier-Stokes equations * Kalman filter, an approximating algorithm in optimal control applications and problems Device * Filter ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu] Cover (topology) In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Bing]   [Yahoo]   [DuckDuckGo]   [Baidu]