Publication date:
7 September 2017
Source:Advances in Mathematics, Volume 317
Author(s): Antongiulio Fornasiero, Philipp Hieronymi, Erik Walsberg
A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every ${\mathbb{R}}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A\subseteq {\mathbb{R}}^k$ is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every ${\mathbb{R}}^n$ 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.