 But combinations of logics also work from another perspective: instead of directly combining logic systems and looking for the interpretation of the resulting system, one can start from a purely mathematical perspective. In van Benthem et al. 2006, for instance, the authors introduce the notion of horizontal and vertical topologies on the product of topological spaces, and show that the modal logic of products of topological spaces with horizontal and vertical topologies coincides with the fusion of S4 with itself. The resulting completeness proof has deep connections with some topological properties of the real and of the rational numbers.