Compactness and local compactness for $${\mathbb{R}}$$ -trees

Abstract

We establish (geometric) criteria for an R-tree to be compact and to be locally compact. It follows that locally compact R-trees are separable. Mathematics Subject Classification (2000). Primary 20E08; Secondary 54E45, 54E50.The purpose of this note is to provide criteria for an R-tree to be compact and to be locally compact. These are expressed entirely in terms of the geometry of the tree. Compact and locally compact R-trees have been used in probability theory; see, for example, [2] and the references cited there. A general reference for the theory of R-trees is [1]. As usual, [x, y] denotes the segment joining points x, y in an R-tree. The ideas of linear subtree, ray and end are discussed in [1, Chapter 2, §4], and for the notions of direction, degree of a point and endpoint see [1] after Lemma 1.7 in Chapter 2.To state the compactness criterion, the following two definitions are needed. Let (X, d) be an R-tree, and let r > 0 be a real number.The second definition uses the idea of projection of a point onto a segment. For a discussion of projection, see [1, Chapter 2, Lemma 1.9] and the remarks following it. The projection of a point z onto a segment [x, y] is simply the point Y(x, y, z) defined after Chapter 2, Lemma 1.2 in [1].