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Prüfer Manifold
In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by and named after Heinz Prüfer. Construction The Prüfer manifold can be constructed as follows . Take an uncountable number of copies ''X''''a'' of the plane, one for each real number ''a'', and take a copy ''H'' of the upper half plane (of pairs (''x'', ''y'') with ''y'' > 0). Then glue the ''open upper half'' of each plane ''X''''a'' to the upper half plane ''H'' by identifying (''x'',''y'')∈''X''''a'' for ''y'' > 0 with the point in ''H''. The resulting quotient space Q is the Prüfer manifold. The images in Q of the points (0,0) of the spaces ''X''''a'' under identification form an uncountable discrete subset. See also *Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves loc ... [...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 separati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analytic
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact Space
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. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory. In 1915 he began his University studies in Mathematics, Physics and Chemistry in Berlin. After that he started his Doctorate degree with Issai Schur as his advisor at Friedrich Wilhelm University, Berlin. In 1921 he obtained his Doctorate degree. His thesis was named ''Unendliche Abelsche Gruppen von Elementen endlicher Ordnung'' (Infinite abelian groups of elements of finite order). This thesis set the road for his contributions on abelian groups. In 1922 he worked with mathematician Paul Koebe in the University of Jena, and in 1923 he obtained tenure and was at this University until 1927. In that year he moved to Münster University where he worked until the end of his life. His final work was about projective geometry, but it was posth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Line (topology)
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments. Definition The closed long ray L is defined as the cartesian product of the First uncountable ordinal, first uncountable ordinal \omega_1 with the Interval (mathematics), half-open interval [0, 1), equipped with the order topology that arises from the lexicographical order on \omega_1 \times [0,1). The open long ray is obtained from the closed long ray by removing the smallest element (0, 0). The long line is obtained by putting together a long ray i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Spaces
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
