Another Characterization of Trees

It is proved that a continuum is a tree if and only if each two connected subsets meet in a connected set. In this note a continuum is a compact connected Hausdorff space. A tree is a continuum in which distinct points are never conjugate [6], i.e., any two distinct points are separated by the deletion of some third point. A dendrite is a metrizable tree. The literature is replete with characterizations of dendrites, but it is frequently the case that these characterizations are invalid for trees. Recently the author [5] has observed that if F is a property which characterizes dendrites among metrizable continua, then it is often the case that an arbitrary continuum is a tree if and only if it satisfies P hereditarily, i.e., if every subcontinuum satisfies P. Therefore one can expect that a property which characterizes dendrites among metrizable continua, and which is intrinsically hereditary, will also characterize trees. To this end let us consider the "connected intersection property." A connected space satisfies the connected intersection property if the intersection of any two of its connected sets is connected. It is obvious that this property is hereditary, and it is known [6] that a metrizable continuum is a dendrite if and only if it satisfies the connected intersection property. The author knows of no published proof which is not highly metric in nature, i.e., in which the second countability of the continuum is employed in a crucial way so that there is no hope of a direct generalization to arbitrary continua. It is the purpose of this note to provide a new proof, valid in the nonmetrizable case. It should be noted that the connected intersection property has been studied in general connected spaces by Whyburn [7] and Kok [3], especially in the locally connected case. Whyburn first proved that in a locally connected, connected Hausdorff space distinct points are never conjugate if and only if the space satisfies the connected intersection property. It is curious that the compact, nonlocally connected case has not been considered. The main difficulty Received by the editors August 21, 1989 and, in revised form, February 22, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20, 54F50, 54F65.