Metric Space

A metric space induces Topological Properties like Open and Closed Set s which leads to the study of even more abstract Topological Space s. HISTORY 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 a 2-tuple (''X'',''d'') where ''X'' is a Set and ''d'' is a Metric on ''X'', that is, a function d such that # ''d''(''x'', ''y'') ≥ 0 (''non-negativity'') # ''d''(''x'', ''y'') = 0 if and only if ''x'' = ''y'' (''identity of indiscernibles'') # ''d''(''x'', ''y'') = ''d''(''y'', ''x'') (''symmetry'') # ''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 ''X'' for a metric space if it is clear from the context what metric is used. If we relax the second condition to allow zero distance between two distinct points, then the space is known as semi-metric space or pseudo-metric space. EXAMPLES
	Any	"http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/normed_vector_space" class="copylinks">Normed Vector Space is a metric space by defining ''d''(''x'', ''y'') = ''y'' &minus ''x'', see also Distances Based On Norms (If such a space is complete, we call it a Banach Space ) Example:
	The	"http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/British_Rail" class="copylinks">British Rail metric (also called the Post Office metric) on a Normed Vector Space , given by ''d''(''x'',&nbsp''y'')=''x''&nbsp+&nbsp''y'' for distinct points ''x'' and ''y'', and ''d''(''x'', ''x'') = 0 The name alludes to the tendency of railway journeys (or letters) to proceed via London , which is identified with the origin However, this comparison breaks down when considering the distance between two stations on the same line

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Metric Space