The modal logic of $${\beta(\mathbb{N})}$$

Let $${\beta(\mathbb{N})}$$ denote the Stone–Cech compactification of the set $${\mathbb{N}}$$ of natural numbers (with the discrete topology), and let $${\mathbb{N}^\ast}$$ denote the remainder $${\beta(\mathbb{N})-\mathbb{N}}$$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of $${\mathbb{N}^\ast}$$ is S4 and that the modal logic of $${\beta(\mathbb{N})}$$ is S4.1.2.