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Nagata–Smirnov Metrization Theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, -locally finite) basis. A topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. A collection in a space X is countably locally finite (or -locally finite) if it is the union of a countable family of locally finite collections of subsets of X. Unlike Urysohn's metrization theorem 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) ..., which provides only a sufficient condition for metrizability, this theorem provides both a necess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...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 topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
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 H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Locally Finite
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields. Examples and properties A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of \mathbb of the form (n, n+2) for an integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of \mathbb of the form (-n, n) for a natural number ''n''. If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urysohn's Metrization Theorem
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] [Baidu] |
Jun-iti Nagata
was a Japanese mathematician specializing in topology. In 1956, Jun-iti Nagata earned his PhD from Osaka University under the direction of Kiiti Morita. He was the author of two standard graduate texts in topology: ''Modern Dimension Theory'' and ''Modern General Topology''. His name is attached to the Nagata–Smirnov metrization theorem The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (t ..., which was proved independently by Nagata in 1950 and by Smirnov in 1951, as well as the Assouad–Nagata dimension of a metric space, which he introduced in a 1958 article. Nagata became a professor emeritus at both Osaka Kyoiku University, where he taught for 10 years, and Osaka Electro-Communication University, where he taught for 5 years. Works * Jun-iti Nagata: ''Modern Dimension Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Mikhailovich Smirnov
Yuri Mikhailovich Smirnov (Юрий Михайлович Смирнов, September 19, 1921, Kaluga – September 3, 2007, Moscow) was a Soviet and Russian mathematician, specializing in topology. Biography Yuri M. Smirnov was born in a family of clerical employees. His mother was imprisoned in 1937 for anti-Soviet activity and, as later revealed, was executed by gun shot. While studying at school, Yuri M. Smirnov was interested in mathematics and astronomy and after completing undergraduate study in 1939 entered the astronomy department of the Faculty of Mechanics and Mathematics of Moscow State University. However, soon under the influence of A. N. Kolmogorov, he transferred to the mathematical department of the same Faculty. After his second year of undergraduate study, Smirnov went in autumn 1941 to the front and served as a radio operator in the Northern Fleet until the end of WW II. After demobilization in 1945, he continued his studies at the Faculty of Mechanics and Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
