On Urysohn's \(\mathbb{R}\)-tree
The author gives a short survey of properties of \(\mathbb{R}\)-trees and, in particular, of the Urysohn \(\mathbb{R}\)-tree. Recall that \(\mathbb{R}\)-trees can be characterized as uniquely geodesic metric spaces in which, for any three different points, the three shortest arcs joining pairs of these points have a unique intersection point. The Urysohn tree \(R\) is an \(\mathbb{R}\)-tree such that \[ R = \bigcup_{n \in \{0\} \cup \mathbb{N}} R_n, \] where \(R_0 = \{0\}\) and \(R_{n+1}\) is obtained from \(R_n\) by gluing to each \(x_n \in R_n\) a space \(C_{x_n}\) which is the union of a continual family of closed half-lines \(\mathbb{R}_{+}\) intersecting pairwise only in zero and such that \(C_{x_n} \cap R_n = \{x_n\}\) and \(C_{x_n} \cap C_{y_n} = \varnothing\) if \(y_n \in R_n\) and \(y_n \neq x_n\). Every pair of points \(x\), \(y \in R\) can be joined in \(R\) by a unique topological segment and the length of this segment is, by definition, the distance between \(x\) and \(y\). Several new interesting results describing the boundary of \(\mathbb{R}\)-trees are obtained. Let \(X\) be a \(\mathbb{R}\)-tree. A boundary of \(X\) is the set of all points \(x \in X\) for which the set \(X \setminus \{x\}\) is connected. Theorem. Let \(Y = \{y_1, \ldots, y_n\}\), \(n \geqslant 4\), be a metric space. If \(Y\) is the boundary of some \(\mathbb{R}\)-tree \(X\), then any point in \(Y\) does not lie between the two other points from \(Y\). If the last condition is satisfied, then there exists at most one minimal \(\mathbb{R}\)-tree \(X\) with boundary \(Y\). The constructive proof of this result is based on \textit{P. Buneman}'s four-point property [J. Comb. Theory, Ser. B 17, 48--50 (1974; Zbl 0286.05102)].
scientific article; zbMATH DE number 7079955
scientific article