Abstract

It is a celebrated result of McKinsey and Tarski [28] thatS4is the logic of the closure algebraΧ+over any dense-in-itself separable metrizable space. In particular,S4is the logic of the closure algebra over the realsR, the rationalsQ, or the Cantor spaceC. By [5], each logic aboveS4that has the finite model property is the logic of a subalgebra ofQ+, as well as the logic of a subalgebra ofC+. This is no longer true forR, and the main result of [5] states that each connected logic aboveS4with the finite model property is the logic of a subalgebra of the closure algebraR+.In this paper we extend these results to all logics aboveS4. Namely, for a normal modal logicL, we prove that the following conditions are equivalent: (i)Lis aboveS4, (ii)Lis the logic of a subalgebra ofQ+, (iii)Lis the logic of a subalgebra ofC+. We introduce the concept of a well-connected logic aboveS4and prove that the following conditions are equivalent: (i)Lis a well-connected logic, (ii)Lis the logic of a subalgebra of the closure algebra$\xi _2^ + $over the infinite binary tree, (iii)Lis the logic of a subalgebra of the closure algebra${\bf{L}}_2^ + $over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logicLaboveS4is connected iffLis the logic of a subalgebra ofR+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics forS4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames forS4and descriptive spaces are isomorphic. It follows that every logic aboveS4is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of bothCandR. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients ofL2is an interior image ofR.