Points very close to the smallest ideal of \(\beta S\) (Q1333175)

For an infinite cardinal \(\kappa\), the \(\kappa\)-topology of a topological space is the topology generated by all intersections of \(\kappa\) (or fewer) open sets in the original topology. The authors consider such topologies on the Stone-Čech compactification \(\beta S\) of a discrete semigroup \(S\). The ordinary closure of the smallest ideal \(K\) of \(\beta S\) may not be an ideal, but the authors show that the \(\kappa\)-closure always is. In the special case when \(S = (\mathbb{N}, +)\) and \(\kappa = \omega\), it is proved that left cancellation fails at each point of \(\omega - \text{cl }K\), and that \(\omega - \text{cl }K \subseteq \mathbb{N}^* + \mathbb{N}^*\). The key to the authors' work is a characterization of those families \(\mathcal A\) of subsets of \(S\) for which there is \(p \in K\) such that \(\text{cl }A\) is a neighborhood of \(p\) for every \(A \in {\mathcal A}\). This characterization is rather technical, but is related to the piecewise syndetic property of subsets in the case \(S = \mathbb{N}\).