The real tree

Abstract

In this note, it is shown that there exists a natural metric on the set T:= {t ⊆ ℝ | sup t < + ∞, inf t ∈ t} of bounded subsets of ℝ containing their infimum which endows T with the structure of an ℝ-tree so that, for every t ∈ T, the "number" of connected components of T\{t} coincides with the cardinality # p(ℝ) of the set of subsets of ℝ. In addition, the set of ends of T is explicitly determined, and various further features of T are discussed, too. © 1996 Academic Press, Inc.