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Regular Topological 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'' have 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] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics 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, Torsion (mechanics), 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 List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of The Separation Axioms
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, there were many definitions offered, some of which assumed (what we now think of as) some separation axioms. For example, the definition given by Felix Hausdorff in 1914 is equivalent to the modern definition plus the Hausdorff space, Hausdorff separation axiom. The separation axioms, as a group, became important in the study of metrisability: the question of which topological spaces can be given the structure (mathematics), structure of a metric space. Metric spaces satisfy all of the separation axioms; but in fact, studying spaces that satisfy only ''some'' axioms helps build up to the notion of full metrisability. The separation axioms that were first studied together in this way were the axioms for accessible spaces, Hausdorff spaces, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-dimensional Space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical illustration of a zero-dimensional space is a point. Definition Specifically: * A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement that is a cover by disjoint open sets. * A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement. * A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets. The three notions above agree for separable, metrisable spaces. Properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 is any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Paul Urysohn had used the notion of completely regular space in a 1925 paper without giving it a name. But it was Andrey Tychonoff who introduced the terminology ''completely regular'' in 1930. 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 such that f(x)=1 and f\vert_ = 0. (Equivalently one can choose any two values instead of 0 and 1 and even require that f be a bounded function.) A to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Compactness
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompactness
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracomp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudonormal Space
In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Note the following: * Every normal space is pseudonormal. * Every pseudonormal space is regular. An example of a pseudonormal Moore space that is not 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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ... was given by , in connection with the conjecture that all normal Moore spaces are metrizable.. References Topology Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Space
Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * Normal (2009 film), ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * ''Normal!'', a 2011 Algerian film * The Normals (film), ''The Normals'' (film), a 2012 American comedy film * Normal (New Girl), "Normal" (''New Girl''), an episode of the TV series Mathematics * Normal (geometry), an object such as a line or vector that is perpendicular to a given object * Normal basis (of a Galois extension), used heavily in cryptography * Normal bundle * Normal cone, of a subscheme in algebraic geometry * Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) * Normal distribution, the Gaussian continuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Quotient
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 indistinguishabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T1 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Hausdorff Space
Completely may refer to: * ''Completely'' (Diamond Rio album) * ''Completely'' (Christian Bautista album), 2005 * "Completely", a song by American singer and songwriter Michael Bolton * "Completely", a song by Serial Joe from ''(Last Chance) At the Romance Dance...'', 2001 * "Completely", a song by Shane Filan from '' Love Always'', 2017 * "Completely", a song by Blue October from '' This Is What I Live For'', 2020 See also * Completeness (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
