On indecomposability of $\beta X$

Abstract

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 $\beta X$ to be indecomposable. We show that indecomposability and irreducibility are equivalent properties in compactifications of indecomposable 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.

12 pages, 5 figures

References in corpus (3)

• Widely-connected sets in the bucket-handle continuum
• Embedding irreducible connected sets
• One-to-one composant mappings of $[0,\infty )$ and $(-\infty ,\infty )$

Cited by in corpus (3)

• Singularities of meager composants and filament composants
• Separable Indecomposable Continuum with Exactly One Composant
• Compactification of cut-point spaces