Arguably the most basic logic for spatial reasoning is S4 interpreted over topological spaces. Our primary goal in this paper is to study a minimal augmentation of this logic. We will study multimodal languages of products of topologies where the modality of each component of the product is preserved into the product. This work generalizes some of the results of D.M. Gabbay and V.B. Shehtman on products of modal logics to the topological setting.
Thus in the most general case, each component modality will have S4 as a complete axiomatization. The interesting question, of course, is how the modalities of various dimensions interact. The expansion of the language enables us to `track dimensions' in the product and this in turn will add some expressive power to the language.