On an example of Simon (Q1348245)

The goal of this paper is to provide a method for constructing compactifications \(\alpha\mathbb{N}\) of the positive integers \(\mathbb{N}\) for which the remainder \(\alpha\mathbb{N}-\mathbb{N}\) is sequentially compact and in which no nontrivial sequence of \(\mathbb{N}\) converges. In addition, it is desired that some control be obtained over the other properties of \(\alpha \mathbb{N}-\mathbb{N}\). The authors note Petr Simon previously exhibited such an example in which \(\alpha\mathbb{N} -\mathbb{N}\) is 0-dimensional. In the current paper, the authors' technique is used to provide an example with \(\alpha\mathbb{N} -\mathbb{N}\) connected. Their method is also used to reproduce Simon's example.