A real tree, or an -tree, is a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 (M,d) such that
for any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 f from an interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [a,b] to M such that f(a)=x and f(b)=y. The condition that the arc is a geodesic segment means that the map f above can be chosen to be an isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

, that is it can be chosen so that for every z, t in [a,b] we have d(f(z), f(t))=|z-t| and that f(a)=x, f(b)=y.

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 space

Injective metric space

In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...

s .

There is a theory of group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

s on R-trees, known as the Rips machine

Rips machine

In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991....

, which is part of geometric group theory

Geometric group theory

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...

.

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

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

 and closed

Closed

Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...

.

Examples

• Each discrete tree
  Tree (graph theory)
  In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
  
   can be regarded as an R-tree by a simple construction such that neighboring vertices have distance one.
• The Paris metric makes the plane into an R-tree. If two points are on the same ray in the plane, their distance is defined as the Euclidean distance
  Euclidean distance
  In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
  
  . Otherwise, their distance is defined to be the sum of the Euclidian distances of these two points to the origin. More generally any hedgehog space
  Hedgehog space
  In mathematics, a hedgehog space is a topological space, consisting of a set of spines joined at a point.For any cardinal number K, the K-hedgehog space is formed by taking the disjoint union of K real unit intervals identified at the origin...
  
   is an example of a real tree.
• The R-tree obtained in the following way is nonsimplicial. Start with the interval [0,2] and glue, for each positive integer n, an interval of length 1/n to the point 1−1/n in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this R-tree. Gluing an interval to 1 would result in a closed
  Closed
  Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
  
  set of singular points at the expense of discreteness.