Indecomposable homogeneous plane continua are hereditarily indecomposable

Abstract

ABSTRACT. F. Burton Jones [7] proved that every decomposable homogeneous plane continuum is either a simple closed curve or a circle of homogeneous nonseparating plane continua. Recently the author [5] showed that no subcontinuum of an indecomposable homogeneous plane continuum is hereditarily decomposable. It follows from these results that every homogeneous plane continuum that has a hereditarily decomposable subcontinuum is a simple closed curve. In this paper we prove that no subcontinuum of an indecomposable homogeneous plane continuum is decomposable. Consequently every homogeneous nonseparating plane continuum is hereditarily indecomposable. 1. Definitions. A space is homogeneous if for each pair p, q of its points there exists a homeomorphism of the space onto itself that takes p to q.A continuum is a nondegenerate compact connected metric space. A continuum is of type A' provided it is irreducible between a pair of its points and admits a monotone upper semicontinuous decomposition, each of whose elements has void interior, whose quotient space is an arc [14].A finite collection {L¡: 1 < z < zz} of open sets is a chain provided that L¡ nLj-^0 if and only if \i-j\ < 1.Throughout this paper R2 is the Cartesian plane with metric p. The closed interval with end points p and q in R2 is denoted by <p, q). We shall denote the closure and the boundary of a given set H by Cl H and Bd H respectively. The union of a collection H of sets is denoted by St H. For definitions of unfamiliar terms and phrases see [5], [9], and [13].

Decompositions of homogeneous continua

“…In § 2, we state the general (relative) version of the theory and indicate how a result of Hagopian [6] may be viewed as a consequence of it.…”

No homogeneous tree-like continuum contains an arc

“…Let 6, be the chain in 5" that goes from x, to x2. Since 5" is a tree chain, (10) each link of 6, intersect 7.…”

A complete classification of homogeneous plane continua

“…A continuum is hereditarily decomposable (hereditarily indecomposable) if every non-degenerate subcontinuum is decomposable (indecomposable, respectively). Hagopian [11] showed that the answer to the question of Knaster and Kuratowski is still yes if the continuum merely contains a hereditarily decomposable subcontinuum. Problème 2 by Knaster and Kuratowski was formally solved by Bing [2] who showed in 1948 that the pseudo-arc, described in detail in Section 1.1, is another homogeneous plane continuum.…”