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实树.
数学上,实树,也称为R-树,是指有类似于树的性质的度量空间(M,d),:对M中任何两点x, y,都有唯一的自x至y的弧,而这条弧是测地线。自x至y的弧,是指从区间[a, b]到M中的拓扑嵌入f,使得f(a)=x,f(b)=y。
一个测地度量空间是实树,当且仅当这空间是δ-双曲空间,且δ=0。
完备实树是单射度量空间。(Kirk 1998)
研究实树上的群作用的理论称为Rips machine,是几何群论的一部分。
一个单纯实树是没有某种奇怪的拓扑性质的实树。实树T中的一点x称为寻常的,意思是T−x有正好两个分支。不是寻常的点x称为奇异的。实树称为单纯的,如果奇异点的集合是离散和闭的。
- 任何离散树都可以视为实树,一个简单的构造方法,是定义相邻节点的距离为1。
- 在平面上定义度量为:平面上两点如果在自原点出发的同一条射线上,则这两点距离为两点的欧几里得距离,若否,则距离为自原点至这两点的欧几里得距离的和。上述度量称为巴黎度量。巴黎度量使平面成为实树。更一般而言,刺猬空间都是实树。
- 下述所构造出的是非单纯实树:取闭区间[0,2],对每个正整数n,把一个长为1/n的区间黏合到原来区间的点1-1/n上。这个实树的奇异点集是离散的,却非闭集,因为在这实树中1是寻常点。如果再黏贴一个区间到点1上,可以使奇异点集成为闭集,但是失了离散性。
- Bestvina, Mladen, ℝ-trees in topology, geometry, and group theory, Handbook of geometric topology, Amsterdam: North-Holland: 55–91, 2002 [2013-12-22], MR 1886668, (原始内容存档于2016-03-03) .
- Chiswell, Ian, Introduction to Λ-trees, River Edge, NJ: World Scientific Publishing Co. Inc., 2001, ISBN 981-02-4386-3, MR 1851337 .
- Kirk, W. A., Hyperconvexity of R-trees (PDF), Fundamenta Mathematicae, 1998, 156 (1): 67–72 [2013-12-22], MR 1610559, (原始内容存档 (PDF)于2016-03-04) .
- Shalen, Peter B., Dendrology of groups: an introduction, Gersten, S. M. (编), Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer-Verlag: 265–319, 1987, ISBN 978-0-387-96618-2, MR 0919830 .
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