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In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 1964). They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that ''X'' and ''Y'' are metric spaces and ƒ is a function from ''X'' to ''Y''. Thus we have a metric map when, for any points ''x'' and ''y'' in ''X'', : Here ''d''''X'' and ''d''''Y'' denote the metrics on ''X'' and ''Y'' respectively. ==Category of metric maps== A map ƒ between metric spaces is an isometry if and only if 1) it is metric, 2) it is a bijection, and 3) its inverse is also metric. The composite of metric maps is also metric. Thus metric spaces and metric maps form a category Met; Met is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphisms in Met are the isometries. 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 map」の詳細全文を読む スポンサード リンク
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