How to avoid a compact set

Antongiulio Fornasiero, Philipp Hieronymi, Erik Walsberg

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)758-785
Number of pages28
JournalAdvances in Mathematics
Volume317
DOIs
StatePublished - Sep 7 2017

Keywords

  • Expansions of the real ordered additive group
  • Hausdorff dimension
  • Marstrand projection theorem
  • Packing dimension
  • Tame geometry
  • Topological dimension

ASJC Scopus subject areas

  • Mathematics(all)

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