The topology of continua that are approximated by disjoint subcontinua (Q1004040)

The term continuum as used here means a compact, connected, metric space. Suppose that \(X\) is a continuum and there exists a one-to-one map \(f:[0,\infty)\to X\) with the property that \(X=\bigcap\{\overline{f([n,\infty))}\,|\,n=0,1,\dots\}\). \textit{S. Curry} [Topology Appl. 39, No. 2, 145--151 (1991; Zbl 0718.54042)], showed that if \(X\subset\mathbb R^2\) and \(\mathbb R^2\setminus X\) has only a finite number of components, then \(X\) is indecomposable. The main result of this paper is a generalization. Theorem 1. Suppose that \(\{Y_i\,|\,i\in\mathbb N\}\) is a pairwise disjoint collection of subcontinua of a continuum \(X\) such that \(\roman{lim}_{i\rightarrow\infty}d_H(Y_i,X)=0\) where \(d_H\) is the Hausdorff metric. Then the following are true. (1) \(X\) is non-Suslinian. (2) If each \(Y_i\) is chainable and \(X\) is finitely cyclic, then \(X\) is indecomposable or the union of two indecomposable subcontinua. (3) If \(X\) is \(G\)-like, then \(X\) is indecomposable. (4) If all of \(\{Y_i\,|\,i\in\mathbb N\}\) lie in the same ray and \(X\) is finitely cyclic, then \(X\) is indecomposable. Most of the terms used in this theorem are defined in the paper.