How to avoid a compact set

Abstract A first-order expansion of the R -vector space structure on R does not define every compact subset of every R n if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ R k is closed and the Hausdorff dimension of A exceeds the topological dimension of A , then every compact subset of every R n can be constructed from A using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension.