The expressive power of first-order topological languages

We consider the expressive power of the first-order structure 〈Ω, C〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation “Region x is in contact with region y.” We prove two main theorems: • Let P[Q] be the domain of bounded, non-empty, rational polyhedra in twoor threedimensional Euclidean space. A relation Γ over P[Q] is definable in the structure 〈P[Q], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions. • Let R be the space of bounded, non-empty, closed regular regions. in n-dimensional Euclidean space. Any analytical relation over lower-dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R, C〉.