On indecomposable subcontinua of β[0,∞) — [0,∞)

Abstract

It is well known that β[0,∞) —[0,∞) is an indecomposable continuum [1]. Van Douwen announced in [7] that there are at least two nonhomeomorphic indecomposable subcontinua in this space. We prove that infinitely many nonhomeomorphic indecomposable subcontinua can be produced by adding Cohen reals. This gives an answer to [6, Question 4]. Our method is as follows. First we prove that every layer, introduced by Mioduszewski in [4], is an indecomposable subcontinuum (Theorem 2.6). Secondly, we prove that if M is a model of ZFC in which c is strongly regular and κ ⩽ c is an uncountable regular cardinal, there is a layer T such that for any λ < κ every nonempty G λ-set of T has nonempty interior and there is a point in T with character κ in the model obtained by adding κ Cohen reals to M . Our Theorems 2.3 and 2.6 answer the problem in [4]. In Section 1, we will discuss some special points in β[0,∞) - [0,∞).