Wikipedia Metric Spaces - Metric space - Wikipedia, the...

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Metric space From Wikipedia, the free encyclopedia In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3- dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line connecting them. The geometric properties of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity. A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces. History Contents 1 History 2 Definition 3 Examples of metric spaces 4 Open and closed sets, topology and convergence 5 Types of metric spaces 5.1 Complete spaces 5.2 Bounded and totally bounded spaces 5.3 Compact spaces 5.4 Locally compact and proper spaces 5.5 Connectedness 5.6 Separable spaces 6 Types of maps between metric spaces 6.1 Continuous maps 6.2 Uniformly continuous maps 6.3 Lipschitz-continuous maps 6.4 Isometries 7 Notions of metric space equivalence 8 Topological properties 9 Distance between points and sets; Hausdorff distance and Gromov metric 10 Product metric spaces 10.1 Continuity of distance 11 Quotient metric spaces 12 See also 13 Notes 14 Sources 15 External links 페이지 1/ ±10 Metric space - Wikipedia, the free encyclopedia 2009-02-14
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Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel , Rendic. Circ. Mat. Palermo 22 (1906) 1–74. Definition A metric space is an ordered pair ( M , d ) where M is a set and d is a metric on M , that is, a function such that for any x , y and z in M 1. d ( x , y ) 0 ( non-negativity ) 2. d ( x , y ) = 0 if and only if x = y ( identity of indiscernibles ) 3. d ( x , y ) = d ( y , x ) ( symmetry ) 4. d ( x , z ) d ( x , y ) + d ( y , z ) ( triangle inequality ). The function d is also called distance function or simply distance . Often d is omitted and one just writes M for a metric space if it is clear from the context what metric is used. Relaxing the second requirement, or removing the third or fourth, leads to the concepts of a pseudometric space, a quasimetric space, or a semimetric space. If the function takes values in the extended real number line, but otherwise satisfies above conditions, then it is called an extended metric ; the corresponding space is then called an -metric space . The first of these four conditions actually follows from the other three, since:
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.

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Wikipedia Metric Spaces - Metric space - Wikipedia, the...

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