  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.
