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

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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.