## The topological product of S4 and S5

The most obvious bimodal logic generated from unimodal logics L_{1} and L_{2} is their fusion, L_{1} ⊗ L_{2}, axiomatized by the theorems of L_{1} for
⬛_{1} and of L_{1} for ⬛_{2}, and by the rules of modus ponens, substitution and necessitation for ⬛_{1} and for ⬛_{2}. Shehtman introduced the frame product L_{1} × L_{2}, as the logic of the products of certain Kripke frames.
Typically, L_{1} ⊗ L_{2} ⊊ L_{1} × L_{2}, e.g. S4 ⊗ S4 ⊊ S4 × S4. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman's idea and introduced the topological product L_{1} ×_{t} L_{2}, 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.