\(\beta\) (X-\(\{\) x\(\}\) ) for X not locally connected
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.
scientific article; zbMATH DE number 4007229