ON A CHARACTERIZATION OF TREES

We prove that a continuum X is a tree if and only if for each pair ici of its nondegenerate subcontinua some subcontinuum of K separates L. The purpose of this note is to give a slight generalization and a somewhat simpler proof of a recent characterization of trees by Ward (1). Theorem. A continuum X is a tree if and only if for each pair of nondegener- ate subcontinua K and L of X such that K c L some subcontinuum of K separates L. A continuum is a compact connected Hausdorff space. A continuum X is a tree if every pair of distinct points of X can be separated by a third point of X. A continuum K in a space X is said to be a continuum of convergence of X if there is a net {Ka : a € X} of subcontinua of X converging to K and such that KnK = 0 and either K = Kr or K" n K = 0 for all a.iel. It is well known and easy to show that a continuum X that contains no continuum of convergence is locally connected. The continuum X is said to be unicoherent if whenever X = K U L, where K and L are subcontinua of X, then K n L is connected. We say that X is hereditarily unicoherent if each of its subcontinua is unicoherent. Proof of the theorem. Assume that X is a tree. Let K c L be nondegenerate subcontinua of X and let x and y be two points of K. If z € X separates x and y then z€K. In particular, {z} separates L. Now assume that for each pair K c L of nondegenerate subcontinua of X some subcontinuum of K separates L. We prove first that X is hereditarily unicoherent. Suppose that K and L are subcontinua of X such that K n L = A u B, where A and B are non-void disjoint closed sets. Let M be a subcontinuum of IT U L which is irreducible with respect to