Archive for Mathematical Logic 48 (3-4):231-242 (2009)

Abstract
Let ${\beta(\mathbb{N})}$ denote the Stone–Čech 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
Keywords Modal logic  Topology  Stone–Čech compactification
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DOI 10.1007/s00153-009-0123-9
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Completeness of S4 with Respect to the Real Line: Revisited.Guram Bezhanishvili & Mai Gehrke - 2004 - Annals of Pure and Applied Logic 131 (1-3):287-301.

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