On indecomposable subcontinua of \(\beta{}[0,\infty{})-[0,\infty{})\)
The remainder in the Stone-Čech compactification \(\beta(N\times I)\), where \(N\) is the set of positive integers and \(I\) is the closed (unit) interval of reals, \(0\leq t\leq 1\), both with usual topologies, is a union of continua \(M\) indexed by points of the remainder in the Stone-Čech compactification \(\beta N\), which are irreducible between points in \(\beta(N\times \{0\})\) and \(\beta(N\times \{1\})\) and have decompositions into layers with the quotient spaces being Hausdorff arcs. The author shows that all the layers in continua \(M\) are indecomposable and that among remote layers there exist non-degenerate ones. This answers questions of the reviewer [Topology and Measure, Part 2, Conf. Zinnowitz 1974, 257-283 (1978; Zbl 0407.54018)]. Assuming new axioms consistent with ZFC, the author shows that there are infinitely many non- homeomorphic indecomposable continua in the remainder of \(\beta(N\times I)\), in particular, continuum many if there is an inaccessible cardinal.
scientific article; zbMATH DE number 90162