Layers of components of 𝛽([0,1]×𝑁) are indecomposable

Smith, Michel

doi:10.1090/S0002-9939-1992-1093605-4

Layers of components of $\beta ([0,1]\times \textbf {N})$ are indecomposable
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Proc. Amer. Math. Soc. 114 (1992), 1151-1156 Request permission

Abstract:

We examine the structure of certain subcontinua of the Stone-Čech compactification of the reals. Let $N$ denote the integers, let $X = [0,1] \times N$, and let $C$ be a component of ${X^*} = \beta X - X$. It is known that $C$ admits an upper semicontinuous decomposition $G$ into maximal nowhere dense subcontinua of $C$ so that $C/G$ is a Hausdorff arc. The elements of $G$ are called layers. It has been shown that the layers of $C$ that contain limit points of a countable increasing or decreasing sequence of cut points of $C$ are nondegenerate indecomposable continua (various forms of this fact have been proven by Bellamy and Rubin, Mioduszewski, and Smith). We show that all the layers of $C$ are indecomposable.*

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Set theory

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