Euclidean Hierarchy in Modal Logic

For an Euclidean space Rn, let Ln denote the modal logic of chequered subsets of Rn. For every n ≥ 1, we characterize Ln using the more familiar Kripke semantics, thus implying that each Ln is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics Ln form a decreasing chain converging to the logic L∞ of chequered subsets of R∞. As a result, we obtain that L∞ is also a logic over Grz, and that L∞ has the finite model property.