Submission date: 2 December 2016

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.

Mathematics Subject Classification: Primary 03C64, Secondary 03C45, 03E15, 28A05, 28A75, 28A80, 54F45

Keywords and phrases:

Full text arXiv 1612.00785: pdf, ps.