A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A⊆Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn 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.
How to avoid a compact set / Fornasiero, Antongiulio; Hieronymi, Philipp; Walsberg, Erik. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - STAMPA. - 317:(2017), pp. 758-785. [10.1016/j.aim.2017.07.011]
How to avoid a compact set
Fornasiero, Antongiulio;
2017
Abstract
A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A⊆Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn 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.File | Dimensione | Formato | |
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