THE REAL TREE

In this note, it is shown that there exists a natural metric on the se t T:={t subset of or equal to R/sup t < + infinity, inf t is an elemen t of t} of bounded subsets of R containing their infimum which endows T with the structure of an IFS-tree so that, for every t is an element of 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 o f T are discussed, too. (C) 1996 Academic Press, Inc.