Continua which have width zero

Abstract

In a recent paper [7], the author showed that with each tree-like continuum there can be associated a non-negative number called the width of M, and it was shown that the plane, E2, does not contain uncountably many disjoint tree-like continua each having a positive width. This result is used here in establishing some conditions under which a tree-like continuum in E2 has width zero. There exists a treelike continuum, such as one which is the sum of a simple triod T and a ray spiralling around T, that has width zero but one of its subcontinua has a positive width. Some of the theorems presented here give conditions under which a tree-like continuum M has width zero hereditarily; that is, every subcontinuum of M has width zero.2 While such a continuum has a thinness property similar to that of a chainable continuum, there do exist in E2 tree-like continua, as indicated by Anderson [1], which have width zero hereditarily but are not chainable. The question in ?4 of [7] and Roberts' result [II] that every chainable continuum has uncountably many disjoint homeomorphic images in E2 suggest the following questions. If the tree-like continuum M is a subset of E2 and has width zero hereditarily, does there exist a sequence of disjoint continua in E2 converging homeomorphically to M? Does a tree-like continuum in E2 have uncountably many disjoint homeomorphic images in E2 if it has width zero hereditarily?3 These questions are not answered, but their converses are direct corollaries to some theorems in [7]. A tree-like continuum M in E2 has width zero hereditarily either if there exists a sequence of disjoint continua in E2 converging homeomorphically to M [7, Theorem 5] or if M has uncountably many disjoint homeomorphic images in E2 [7, Theorem 10]. In this paper, a compact connected metric space is called a continuum. Definitions of trees, chains, tree-like continua, and triods can be found in [6]. A definition of the width of a tree-like continuum is stated in [7], and the following property follows directly from this definition of width. A tree-like continuum M has width zero if, and