The Topological Product of S4 and S5

The most obvious bimodal logic generated from unimodal logics L1 and L2 is their fusion, L1 ⊗L2, axiomatized by the theorems of L1 for 1 and of L2 for 2, and by the rules of modus ponens, substitution and necessitation for 1 and for 2. Shehtman introduced the frame product L1×L2, as the logic of the products of certain Kripke frames. Typically, L1 ⊗ L2 ( L1 × L2, e.g. S4⊗ S4 ( S4× S4. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product L1 ×t L2, as the logic of the products of certain topological spaces: they showed, in particular, that S4 ×t S4 = S4 ⊗ S4. In this paper, we axiomatize S4 ×t S5, which is strictly between S4 ⊗ S5 and S4 × S5. We also apply our techniques to proving a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.