Homogeneous Suslinian continua (Q2882469)

Fitzpatrick and Lelek showed that for each metric Suslinian continuum \(X\) the set of all points at which it is connected im kleinen is dense in \(X\). The authors extend their result to Hausdorff Suslinian continua.NEWLINENEWLINEA continuum is a connected compact Hausdorff space. A continuum is Suslinian if it possesses only countably many pairwise disjoint non-degenerate subcontinua.NEWLINENEWLINEThe following are the main results of the paper.NEWLINENEWLINE1. Suppose \(X\) is a non-degenerate Suslinian continuum. Then every non-empty open set in \(X\) contains a Cantor set at every point of which \(X\) is connected im kleinen.NEWLINENEWLINE2. A Suslinian non-separable homogeneous continuum is a simple closed curve.NEWLINENEWLINE3. Let \(X\) be a homogeneous Suslinian compactum with \(\dim X>0\). Then \(X\) is the pairwise disjoint union of finitely many simple closed curves. If \(X\) is also separable then it is metrizable.