Topology and Modality: The Topological Interpretation of First-Order Modal Logic

Abstract
As McKinsey and Tarksi showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the "necessity" operation is modeled by taking the interior of an arbitrary subset of a topological space. in this paper the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics
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Philip Kremer (2009). Dynamic Topological S5. Annals of Pure and Applied Logic 160 (1):96-116.
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