Continua in R∗

Abstract

R ∗ is the Stone–Čech remainder of the real line. We prove that every decomposable continuum in R ∗ is a section of a standard continuum. Every indecomposable continuum in R ∗ is the union of a family of standard continua. About the general structures of continua in R ∗, the following statements are true: 1. (1) Let C and D be continua in R ∗. If one of them is indecomposable, then C ⊂ D, D ⊂ C or C ∩ D = Ø. 1. (2) R ∗ is hereditarily unicoherent, i.e., any intersection of continua in R ∗ is a continuum. Moreover, any intersection of indecomposable continua in R ∗ is indecomposable. 1. (3) The closure of the union of a chain of indecomposable continua in R ∗ is an indecomposable continuum. 1. (4) A point x of R ∗ is not a sub cutpoint iff { x} is the intersection of a maximal chain of nondegenerate indecomposable continua in R ∗. 1. (5) There are no Q-points in ω ∗ iff every composant of β[0, ∞)−[0, ∞) is the union of a strictly increasing sequence of proper indecomposable subcontinua. 1. (6) The principal NCF is equivalent to the statement that β[0, ∞)−[0, ∞) is the union of a strictly increasing sequence of proper indecomposable subcontinua. Now we know in ZFC that there are nine different continua in R ∗.