It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.
We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can be described by modal formulas zemn which generalize the well-known Zeman formula zem. We show that the modal logic S4.Zn := S4 + zemn is the basic modal logic of T1-spaces of modal Krull dimension ≤ n, and we construct a countable dense-in-itself ω-resolvable Tychonoff space Zn of modal Krull dimension n such that S4.Zn is complete with respect to Zn. This yields a version of the McKinsey-Tarski theorem for S4.Zn. We also show that no logic in the interval [S4n+1S4.Zn) is complete with respect to any class of T1-spaces.
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