Compactifications of indecomposable topological spaces

Abstract

A continuum is a compact and connected topological space. A continuum that is not the union of any two of its proper subcontinua is said to be indecomposable. We examine topological spaces which are closely related to indecomposable continua, specifically, widely-connected spaces and the hyperspace of the Stone-Cech remainder of the half-line. A widely-connected space is a connected space all of whose non-trivial connected subsets are dense in the entire space. We answer questions of Erdos, Bellamy, and Mioduszewski with the following examples: a completely metrizable widely-connected subset of the plane; a widely-connected subset of Euclidean 3-space that is not indecomposable upon the addition of a single limit point; a widely-connected subset of Euclidean 3-space that is contained in a composant of each of its compactifications; widely-connected spaces with large cardinalities. Then, we construct a maximum possible number of non-homeomorphic subcontinua of C(H*), each of which is a union of two order arcs. We also characterize when beta X is indecomposable and study the structure of C(H*) using the property of Kelley.