# Multimodal Logics of Products of Topologies

J. van Benthem, G. Bezhanishvili, B. ten Cate and D. Sarenac
Studia Logica: An International Journal for Symbolic Logic
Vol. 84, No. 3 (Dec., 2006), pp. 369-392
Stable URL: http://www.jstor.org/stable/20016840
Page Count: 24

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## Abstract

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion ${\bf S4}\oplus {\bf S4}$. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers ${\Bbb Q}\times {\Bbb Q}$ with the appropriate topologies.

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