  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.
