Weak cutpoint ordering on hereditarily unicoherent continua

Abstract

1. Introduction. We present here two criteria for determining when an hereditarily unicoherent continuum admits the structure ot a partially ordered space which is closed, and has a unique minimal element (definitions below). Our interest in such partial orderings stems from the fact (Hunter [1]) that a one-dimensional compact connected semigroup with zero and unit admits such a partial ordering. Ward [5] has studied a class of partially ordered spaces, called generalized trees, which he characterized as hereditarily unicoherent continua which admit a closed, order-dense partial with unique minimal element, We improve his characterization by replacing order dense by the weaker monotone, and we give two other characterizations, one of which is completely topological. 2. Definitions. A partial < on a space X is a relation on X (a subset of X XX) which is reflexive, transitive, and antisymmetric. We assume throughout that X is a Hausdorff space, but not necessarily metrizable. The term arc is used to denote a continuum irreducibly connected between two points. For xCX we set L(x) = {yX:y<x}, and for ACX, L(A)=U{L(x):xCA}. We say that < is monotone if L(x) is connected for each xEX or continuous