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In mathematics, a real tree, or an -tree, is a metric space (''M'',''d'') such that for any ''x'', ''y'' in ''M'' there is a unique arc from ''x'' to ''y''. Here by an ''arc'' from ''x'' to ''y'' we mean the image in ''M'' of a topological embedding ''f'' from an interval () to ''M'' such that ''f''(''a'')=''x'' and ''f''(''b'')=''y'' (for some real numbers ''a'' and ''b''). Note that uniqueness refers to the image in ''M''. Moreover, by choosing ''a'' and ''b'' so that ''d(x, y)''=|''a-b''| and using arclength parametrization for the interval (''b'' ), we may assume that this arc is a geodesic segment. The condition that the arc is a geodesic segment means that the map ''f'' above is an isometric embedding, that is, for every ''z, t'' in () we have ''d(f(z), f(t))''=|''z-t''|. Equivalently, a geodesic metric space ''M'' is a real tree if and only if ''M'' is a δ-hyperbolic space with δ=0. Complete real trees are injective metric spaces . There is a theory of group actions on R-trees, known as the Rips machine, which is part of geometric group theory. == Simplicial R-trees == A simplicial R-tree is an R-tree that is free from certain "topological strangeness". More precisely, a point ''x'' in an R-tree ''T'' is called ordinary if ''T''−''x'' has exactly two components. The points which are not ordinary are singular. We define a simplicial R-tree to be an R-tree whose set of singular points is discrete and closed. 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「Real tree」の詳細全文を読む スポンサード リンク
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