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In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable. In differential geometry, the word "metric" may refer to a bilinear form that may be defined from the tangent vectors of a differentiable manifold onto a scalar, allowing distances along curves to be determined through integration. It is more properly termed a metric tensor. ==Definition== A metric on a set ''X'' is a function (called the ''distance function'' or simply distance) :''d'' : ''X'' × ''X'' → [0,∞), where [0,∞) is the set of non-negative real numbers (because distance can't be negative so we can't use R), and for all ''x'', ''y'', ''z'' in ''X'', the following conditions are satisfied: # ''d''(''x'', ''y'') ≥ 0 (''non-negativity'', or separation axiom) # ''d''(''x'', ''y'') = 0 if and only if ''x'' = ''y'' (coincidence axiom) # ''d''(''x'', ''y'') = ''d''(''y'', ''x'') (''symmetry'') # ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') (''subadditivity'' / ''triangle inequality''). Conditions 1 and 2 together define a ''positive-definite function''. The first condition is implied by the others. A metric is called an ultrametric if it satisfies the following stronger version of the ''triangle inequality'' where points can never fall 'between' other points: : ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')) for all ''x'', ''y'', ''z'' in ''X''. A metric ''d'' on ''X'' is called intrinsic if any two points ''x'' and ''y'' in ''X'' can be joined by a curve with length arbitrarily close to ''d''(''x'', ''y''). For sets on which an addition + : ''X'' × ''X'' → ''X'' is defined, ''d'' is called a translation invariant metric if :''d''(''x'', ''y'') = ''d''(''x'' + ''a'', ''y'' + ''a'') for all ''x'', ''y'' and ''a'' in ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Metric (mathematics)」の詳細全文を読む スポンサード リンク
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