On indecomposability of βX

The following is an open problem in topology: Determine whether the Stone-Čech compactification of a widely-connected space is necessarily an indecomposable continuum. Herein we describe properties of X that are necessary and sufficient in order for βX to be indecomposable. We show that indecomposability and irreducibility are equivalent properties in compactifications of widely-connected separable metric spaces, leading to some equivalent formulations of the open problem. We also construct a widely-connected subset of Euclidean 3-space which is contained in a composant of each of its compactifications. The example answers a question of Jerzy Mioduszewski.