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Non-Hausdorff Manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff. Examples Line with two origins The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line. This is the quotient space of two copies of the real line, \R \times \ and \R \times \ (with a \neq b), obtained by identifying points (x,a) and (x,b) whenever x \neq 0. An equivalent description of the space is to take the real line \R and replace the origin 0 with two origins 0_a and 0_b. The subspace \R\setminus\ retains its usual Euclidean topology. And a local base of open neighborhoods at each origin 0_i is formed by the sets (U\setminus\)\cup\ with U an open neighborhood of 0 in \R. For each origin 0_i the subspace obtained from \R by replacing 0 with 0_i is an open neighborhood of 0_i homeomorphic to \R. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
General Topology
In mathematics, general topology (or point set 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. 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 ''topology''. A set with a topology is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Locally Hausdorff Space
In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology. Examples and sufficient conditions * Every Hausdorff space is locally Hausdorff. * There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space. * The line with two origins is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff. * The etale space for the sheaf of differentiable functions on a differential manifold is not Hausdorff, but it is locally Hausdorff. * Let X be a set given the particular point topology with particular point p. The space X is locally Hausdorff at p, since p is an isolated point in X and the singleton \ is a Hausdorff neighbourhood of p. For any other point x, any neighbourhood of it contains p and therefore the space is not locally Hausdorff at x. Properties A space is locally Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Metrizable Space
In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metric d : X \times X \to [0, \infty) such that the topology induced by d is \tau. ''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 space, Hausdorff paracompact spaces (and hence Normal space, normal and Tychonoff space, Tychonoff) and First-countable space, first-countable. However, some properties of the metric, such as Complete metric space, 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 mapping, contraction maps than a metri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
