On Local Trees

where I denotes the class of dendrites, II is the regular curves, III is the hereditarily locally connected continua, IV is the rational curves, and V» is the class of continua of dimension not exceeding n. Now it should be remarked that, for each of these classes save the first, if a continuum possesses the defining property in the small then it is a member of the class. That is, a local regular curve is a regular curve, a local rational curve is a rational curve, and so on, but a local dendrite is not in general a dendrite. However, a local dendrite is a regular curve (but not conversely) so that between the classes I and II there is a distinct and interesting class of continua which warrants study. So far as the author is aware this class has received scant attention in the literature. In this paper we give three characterizations of local dendrites and trees. We show that a Peano continuum is a local dendrite if and only if it contains no small simple closed curves and if and only if it is an absolute neighborhood retract of dimension not exceeding unity. The third characterization is order-theoretic and is an extension of a similar theorem of the author's for trees [8]. The problem of characterizing spaces in terms of their order properties deserves further comment. In the presence of arcwise connectedness and very strong unicoherence properties the inherent order structure of connected spaces has proved to be an effective tool in characterization theorems [2; 5; 8-11 ]. However, even such a simple class of spaces as the regular curves has offered considerable difficulty and so far has not been tractable to discussion in terms of order. Two results lend encouragement to the task. Wallace [ö] has shown that only the indecomposable continua have truly trivial partial order relations and the author [7] has shown that any locally connected continuum can be partially ordered in terms of its cutpoints. If, in the latter case, a way can be found to "extend" the partial order over the cyclic elements of a locally connected continuum