Topological completeness of extensions of S4

Perhaps the most celebrated topological completeness result in modal logic is the McKinseyTarski theorem that if we interpret modal diamond as topological closure, then S4 is complete for the real line or indeed any dense-in-itself separable metrizable space [10]. This result was proved before relational semantics for modal logic was introduced. In the last 15 years, utilizing relational semantics for S4, a number of different proofs of this result appeared in the literature. Completeness of S4 for the real line can be found in [1, 4, 13, 12], for the rational line in [2, 12], and for the Cantor space in [11, 1]. For a topological space X, let X be the closure algebra of all subsets of X. Then completeness of S4 for the real line R means that S4 is the modal logic of the closure algebra R, and the same is true for the rational line Q and the Cantor space C. In [3], the notion of a connected normal extension of S4 was introduced, and it was shown that each connected normal extension of S4 that has the finite model property (FMP) is the modal logic of a subalgebra of R. It was also shown that each normal extension of S4 that has FMP is the modal logic of a subalgebra of Q, as well as the modal logic of a subalgebra of C. It was left as an open problem [3, p. 306, Open Problem 2] whether a connected normal extension of S4 without FMP is also the modal logic of some subalgebra of R. Our purpose here is to solve this problem affirmatively by showing that each connected normal extension of S4 (with or without FMP) is in fact the modal logic of some subalgebra of R. We also prove that each normal extension of S4 (with or without FMP) is the modal logic of a subalgebra of Q, as well as the modal logic of a subalgebra of C. These results generalize similar results from [3] for normal extensions of S4 with FMP to all normal extensions of S4.