\(\beta\) (X-\(\{\) x\(\}\) ) for X not locally connected
type: Item
instance of: scholarly article
title: \(\beta\) (X-\(\{\) x\(\}\) ) for X not locally connected
zbMATH Open document ID: 0621.54023
DOI: 10.1016/0166-8641(87)90044-7
author: Michel Smith
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: In this paper a continuum is a compact connected Hausdorff space and an indecomposable continuum is a continuum which is not the union of two proper subcontinua. The study of indecomposable continua has produced many surprises. At one time even the existence of such a plane continuum was a surprise. That nonmetric indecomposable continua exist ''in nature'' came as something of a surprise when Bellamy observed that for the nonnegative reals A, \(\beta\) A-A is nonmetric, connected and indecomposable. If one thinks of A as I (the closed interval [0,1] minus \(\{\) \(1\})\), then \(\beta\) (I-\(\{\) \(1\})\)-I is indecomposable. Clearly I is locally connected at \(\{\) \(1\}\) and \(\{\) \(1\}\) is not a local cut point of I. So by contrast, the author shows: If w is point of the metric continuum X, w is not a local cut point of X, and X is not connected im kleinen at w, then \(\beta\) (X-\(\{\) \(w\})\)-X is a decomposable continuum and has only one composant. Without any assumption about the behavior of X at w, \(\beta\) (X-\(\{\) \(w\})\)-X is never hereditarily indecomposable. However, if X is nondegenerate and hereditarily indecomposable, then \(\beta\) (X-\(\{\) \(x\})\)-X contains a hereditarily indecomposable continuum.
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54D40
zbMATH DE Number: 4007229
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: nonmetric indecomposable continua
zbMATH Keywords: hereditarily indecomposable continuum
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90044-7
OpenAlex ID: W2071562681
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description: scientific article; zbMATH DE number 4007229