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arXiv · 1703.06862

On indecomposability of $\beta X$

David Sumner Lipham
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Abstract

The following is an open problem in topology: Determine whether the Stone-\v{C}ech 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.

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David Sumner Lipham. 2017-03-20. On indecomposability of $\beta X$. https://doi.org/10.1016/j.topol.2018.05.003

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