How to avoid a compact set

2016/12/02 by Fornasiero, Antongiulio, Hieronymi, Philipp, Walsberg, Erik
#03E15 #28A05 #28A75 #28A80 #54F45 #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG) #Primary 03C64 #Secondary 03C45

paper · doi:10.48550/arxiv.1612.00785

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 ⊆ ℝ^k 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.

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