The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

Jana Maříková; Erik Walsberg

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arXiv · 1510.07196

The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

Jana Maříková·Erik Walsberg

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Abstract

Let $R$ be an o-minimal expansion of the real field. We show that the Hausdorff dimension of an $R$-definable metric space is an $R$-definable function of the parameters defining the metric space. We also show that the Hausdorff dimension of an $R$-definable metric space is an element of the field of powers of $R$. The proof uses a basic topological dichotomy for definable metric spaces due to the second author, and the work of the first author and Shiota on measure theory over nonarchimedean o-minimal structures.

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BibTeX RIS

Jana Maříková, Erik Walsberg. 2015-10-25. The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field. https://arxiv.org/abs/1510.07196

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