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We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]